Depends on what you mean.
Get to high altitude?
Yes you can.
It is really inefficient to get there, but some aircraft already do it.
Achieve a stable orbit and be able to maintain control?
No, at those altitudes you do not have enough oxygen to keep your engines running, you have to bring your own (and thus use a rocket) For the sake of clarity, some definitions: reaction engine : includes both rockets and jets.
The engine pushes some mass in one direction and by reaction it achieves thrust in the opposite one. jet (engine): is used to refer to airbreathing engines.
The aircraft carries the fuel, but not the oxidizer; that is taken from the surrounding atmosphere.
rocket (engine): is used to refer to non-airbreathing engines.
The (space)craft carries both the fuel and the oxidizer.
No atmosphere is needed.
space :
above 100km of altitude.
go to space : staying in space :
Images from xkcd
aviation.stackexchange.com/users/1467/Federico
For certain values of "go to space", yes.
The Karman line (the altitude at which an airplane cannot generate enough lift to stay aloft at any speed slower than orbital velocity) is usually considered the lower edge of outer space.
This implies that with a powerful engine and enough fuel it's possible to "fly" up into space: you just keep gaining altitude and speed until you find yourself in orbit.
No aircraft has actually done this.
The X-15 and SpaceShipOne were both able to reach that altitude in zoom climbs; neither went fast enough to stay there.
aviation.stackexchange.com/users/1677/Mark
It depends what you include in "non-rocket aircraft".
Wikipedia has a whole article on " Non-rocket spacelaunch " which discusses the various options.
The short version is, there are no current space launch systems which do not use rockets.
The most practical one we know
of appears to be space guns , which have been successfully used for sub-orbital launches
(but are not practical for human spaceflight).
aviation.stackexchange.com/users/414/Colin Pickard
A jet engine can only accept air at around half the speed of sound.
If the plane is flying faster, the inlet system needs to slow the air down before accelerating it again.
Hence the higher the speed, the less thrust it can provide.
It is not like a rocket engine that always pulls with comparable force.
A plane needs to accelerate to Mach 25 or about to fly into orbit as a spacecraft, and this is only possible with a new type of engine.
Scramjet that is a jet engine, not a rocket, could probably do this.
However this engine seems still under development and currently a rocket is required to accelerate to the speed under that this engine can even be started.
aviation.stackexchange.com/users/917/h22
I'm not sure how you'd classify the proposed Skylon spacecraft spacecraft.
It would be an airbreathing reactive engine in lower altitudes but a rocket above 26 km.
It might be a game changer.
But this time I don't give even odds that Skylon will come to pass.
aviation.stackexchange.com/users/2099/HopDavid
I recall hearing about F-15s making ballistic suborbital flights, by simply getting up enough of a head of steam to coast through near-space.
I don't think they actually went into "space" (100 km altitude) and they certainly didn't make orbit.
Does anyone else recall the specifics on this?
I can find some references online to using F-15s to test launch ASAT missiles, but my recollection was that the whole aircraft actually went high enough to get well above maximum cruising altitude.
aviation.stackexchange.com/users/1078/Phil Perry
No aircraft powered by engines which rely on atmospheric oxygen can operate outside of Earths atmosphere, 100,000 ft is approximately the altitude limit for air breathing aircraft using conventional fuels, higher sustained altitudes might be possible using liquid hydrogen, but the utility of very high speed and altitude performance as a means of avoiding interception is questionable today.
No air breathing aircraft is likely to outrun a rocket propelled Surface to Air Missile however apparently impressive it's performance statistics.
aviation.stackexchange.com/users/26711/J. Southworth
I always wondered if it was possible for planes and other aircraft to leave the Earth's atmosphere. Normal commercial transport airplanes can fly pretty high, but then they need air to get the speed boost from. Reactive engines seem like a better idea, but I don't know how much importance air has for them.  Is it possible to get to space from Earth's surface using a non-rocket aircraft?
Can aircraft other than rockets go to space?
aviation.stackexchange.com
<aerodynamics><jet-engine><aircraft-limitations>
aviation.stackexchange.com/questions/3425
aviation.stackexchange.com/users/1542/user1306322
It is not currently possible to get into space using an aircraft. 100,000 feet is the altitude limit for engines relying on atmospheric oxygen. At present, there are no space launch systems that do not use rockets. A plane needs to reach speeds of around Mach 25 to fly into orbit. This is only possible with a new type of engine. There have been reports of F-15s making ballistic suborbital flights simply by picking up enough speed to coast through near-space. There are online references to F-15s being used to test-launch ASAT missiles. However, according to the respondent's recollection, the entire aircraft actually rose high enough to exceed maximum cruising altitude. Aircraft powered by engines that are reliant on atmospheric oxygen cannot function outside of the Earth's atmosphere. The altitude limit for air-breathing aircraft that use conventional fuel is around 100,000 ft. It might be possible for higher altitudes to be reached for prolonged periods using liquid hydrogen, but it is questionable how useful very high speed and altitude performance as a means of avoiding interception is nowadays.
At present, there are no space launch systems that do not use rockets. A plane needs to reach speeds of around Mach 25 to fly into orbit. This is only possible with a new type of engine. There have been reports of F-15s making ballistic suborbital flights simply by picking up enough speed to coast through near-space. There are online references to F-15s being used to test-launch ASAT missiles. However, according to the respondent's recollection, the entire aircraft actually rose high enough to exceed maximum cruising altitude. Aircraft powered by engines that are reliant on atmospheric oxygen cannot function outside of the Earth's atmosphere. The altitude limit for air-breathing aircraft that use conventional fuel is around 100,000 ft. It might be possible for higher altitudes to be reached for prolonged periods using liquid hydrogen, but it is questionable how useful very high speed and altitude performance as a means of avoiding interception is nowadays.