How Many Times Can You Replace Porcelain Veneers?

It sounds like the veneers are "popping" off due to issues with the bonding process or potential occlusion (bite) problems.  Veneers should be able to be  replaced without an issue unless there is not an adequate amount of enamel left to bond to.  For these cases you will be bonding to dentin and it would be wise to transition to all porcelain crowns.  These crowns can look just as good as veneers as they are made of the same material.  If your bite is "off" or you are grinding then you  should consider getting a nightguard to protect the new porcelain crowns.

Every time veneers are removed, a little bit of tooth structure comes with it.  A skilled dentist can minimize the amount removed, but if the first set was placed aggressively (more enamel and dentin than needed), then there is less to work with.

If a skilled cosmetic dentist is saying it is time for crowns, then perhaps they are right.

Conservative veneer placement means that they can be replaced many times.  Aggressive placement means fewer before crowns are needed.

Veneers popping off can mean the "glue" used isn't good enough, or that the bite is off and the bite forces are exceeding the strength of the "glue".  The brand of porcelain is a factor as well, as some porcelains don't bond as easily.

Veneers can be replaced several times so long as the tooth structure underneath is solid.  The only way your dentist would be able to tell is by removing the veneers themselves.  Keep in mind that every time a tooth is worked on, you risk having irreversible damage caused to the tooth. With that said, it's always best to try to keep them as veneers rather than crowns because veneers are more conservative.  It's possible they are "popping off" because your bite is off.  It could also be as a result from grinding/clenching your teeth. My recommendation would be to do some research, find a doctor that is very experienced with doing cosmetic treatment, and then have a consultation to see what can be done for you.

Veneers need to be bonded properly to enamel to ensure longevity. Also your bite needs to be properly adjust so that you do not grind or bite on the veneers with excessive bite pressure.

If there is limited amounts of enamel then I agree that crowns might be a better option.

Your original veneer experience is what you should expect.  The most common cause of veneer failure is chipping or breaking.  Having the restorations coming off should be very rare with proper preparation of the tooth, proper bite adjustment, and excellent adhesive techniques.  To me it is more semantics between veneer or crown.  I use the same porcelain material for both and conserve as much tooth structure as possible without compromising the final result.