Yes - numbness denotes a problem with bike fit or a thermal problem.
Here's my personal experience.
Fingers - riding a MTB with "ergonomic" grips, the sort with a wee wing that rests under your palm.
With these I get numb fourth and fifth fingers on both hands.
It takes a three hour ride to start, and if I go 4+ hours then the numbness can last into the evening.
Feet/toes - I have road shoes that are perhaps too small.
If I wear thick socks by mistake, I can lose feeling in toes, again after a long ride.
Answer there is better-fitting shoes, but the interim fix is thin socks.
Neck - More sore than numb, but a generic helmet often has a brim or visor.
This is fine for a MTB but makes you lift your face more on a road bike, leading to a sore neck.
Unclipping the visor is a cheap and easy alievement.
Backside - an upright posture puts more weight on your saddle, via your bum.
A good saddle holds the sit-bones, and not the softer tissues.
A poor saddle presses the wrong things and makes a ride uncomfortable, and numb.
The fix is a saddle replacement, or temporarily spend more time on your feet/pedals and be less efficient but less sore.
`nads - this one is related to repeated friction inducing chafing more than numbness.
A ride under 20 km doesn't need special clothes generally.
Longer rides you'll appreciate padded pants.
A bad fit will leave somewhere scraping back and forth every pedal revolution.
Personal Story -
A flat singletrack MTB course with minor ups and downs, made my padded pants drop a little so the pad's seam was rubbing on the top of the thigh.
I wore a blister that popped.
Since it was around the corner I couldn't see how bad it was, but the hot shower made it quite obvious.
The fix?
tighter pants and pull them up.
No underwear underneath, and last resort is to apply some kind of friction-reducing cream.
Cold - if you're cold you're going to go numb.
Earlobes, nose, cheeks, fingers/toes, and limbs.
I also notice a cold stomach when I stop.
This is exacerbated by the wet.
Personal story
- I once rode through a deep unexpected puddle of snow-melt and got wet up to mid-calf.
My shoes were light "hard drive bag" open-weave.
And then I had to ride ~20km home through a cold easterly wind.
When I got off the bike, I could not feel one foot at all.
The numb sensation did not go away with a hot shower - it took a full 24 hours to vanish completely.
Thermal generally - be prepared to pull up or roll down your sleeves, unzip your top while riding.
If you have an uphill ahead, drink 2 minutes before the start and open all the vents.
If you have a fast downhill coming, button things so you don't get chilled on the way down while coasting.
Rain - if its predicted to rain, consider not riding, or take the wet-weather bike with mudguards/fenders and treadded tyres instead of the slick roadbike.
Take
/Wear a waterproof outer layer and put it on if it does start to drizzle.
bicycles.stackexchange.com/users/19705/Criggie
I've never been a runner so I can't really comment on similarities, but certainly it is possible to experience numbness while riding.
I have poor circulation anyway, and for me it usually happens on long rides in low temperatures.
Warm weather rides are fine.
Also note that there are various medical conditions (e.g. diabetes) which can give rise to poor circulation, so someone who for example is diabetic may be predisposed to numbness.
bicycles.stackexchange.com/users/4808/PeteH
Not generally.
Cyclists have points of contact - hands, feet, butt - where prolonged pressure is created.
Pressure can cut off communication between nerves and the brain, making a body part or area feel numb.
Something could happen with other areas, like the neck for example, if blood flow is impinged and the nerves don't receive oxygen.
This is why wiggling things helps, keeps blood flowing and nerves firing.
bicycles.stackexchange.com/users/19730/jqning
The only numbness I have ever experienced on long rides (6+ hours) was due to poor fit or equipment failure.
The endurance cyclists I known have never described anything "going numb" as a regular occurrence with distance riding.
I can tell you that at the end of a very long ride (12+ hours)
I am generally starting to get sore.
I have not great knees, and the tendons and ligaments in them tend to become inflamed after more than 12 hours or so of riding.
They hurt, I wish more than anything that they went numb.
There tends to be an odd phenomenon where after many hours on a bike doing anything except sitting or laying down
I find horrible.
Walking and or moving is just uncomfortable, it's preferable to ride more, or not move at all.
bicycles.stackexchange.com/users/13710/Deleted User
Coming from the perspective of a cyclist commuter who also does cycle touring, but sometimes doesn't cycle for months: Parts of your body can go numb, yes, but it's less about how fit you are, and more about how much practise you have had.
For example if you haven't cycled regularly then an hour or two in the saddle can leave you with a numb feeling in your backside where it rests on the saddle.
But if you keep doing regular rides then this will decrease and disappear.
Numbness can also occur in the hands and arms, but tends to be over longer periods of time (more than 2 hours).
As you were asking people who don't bicycle, their lack of practise will result in numb backsides if they don't cycle regularly.
A big factor is also the equipment you use.
For example buy a saddle with a groove down the centre which will prevent your soft parts from becoming numb, buy handlebar grips with flat pads at the end to rest your hands on, because the best place to place pressure on your palm is the outside edge (see Tip#4 http://www.builtlean.com/2011/02/23/how-to-proper-push-up-form/ ).
In my experience having a saddle with a groove and handlebar grips with flat pads makes a huge difference on long rides.
Example saddles: google bike saddle with groove Example handlebar grips: http://www.downtheroad.org/Equipment/Bike_Parts/Bike_handlebars.htm
bicycles.stackexchange.com/users/23881/martagnan
When talking to people who don't bicycle themselves, I've noticed that every time I mention bicycling long distances, people seem to think that your body goes numb, like it does when you run long distances. Does it? Is it in any way related to how fit the individual is?
Do riders bodies go numb after hours of riding?
bicycles.stackexchange.com
<long-distance><distance-riding>
bicycles.stackexchange.com/questions/36143
bicycles.stackexchange.com/users/14636/Hellreaver
Cyclists generally concur that numbness results from bike fit problems, circulation issues, cold weather or being unused to spending long periods of time in the saddle. In addition, medical conditions, such as diabetes, may cause riders to feel numb.  Most obviously a poor saddle will make your butt numb. A saddle with a groove and handlebar grips with flat pads makes a huge difference on long rides. Suitable clothing for the cold weather and rain will stop numbness. Numbness often indicates pressure cutting off blood flow, which can be solved with a better fitting saddle and other bike parts, more padding and properly fitting clothing and shoes. Low circulation can also be caused by medical conditions including diabetes. Low temperatures can also cause or exacerbate numbness, which can be addressed with suitable clothing or limiting riding to warmer temperatures. Riders may experience numbness if they are not used to cycling regularly, but this will improve with practise.
Numbness often indicates pressure cutting off blood flow, which can be solved with a better fitting saddle and other bike parts, more padding and properly fitting clothing and shoes. Low circulation can also be caused by medical conditions including diabetes.  Low temperatures can also cause or exacerbate numbness, which can be addressed with suitable clothing or limiting riding to warmer temperatures. Riders may experience numbness if they are not used to cycling regularly, but this will improve with practise.