As for some snags I've heard about the annulment process, here are a few ones:

First, some dioceses require you to start your petition at the parish level.  You cannot just call up the Tribunal, fill out the paperwork, and send it in.  This is a barrier especially for lapsed Catholics.  Many parishes won't help you with something like an annulment until you register and some lapsed Catholics don't want to register until they are certain they can receive Communion.

Second,  there are bad exes, lack of witnesses, or hostile witnesses.  This is the biggie.  It takes lots of time and effort to go through an annulment and not everyone is willing to help.  This is especially true for non-Catholics.

Third, wide variations in the amount of paperwork, fees, and customer service.  For instance, there are some dioceses that assign people advocates they can call while others only allow them to follow up on their petition in writing.