Painting - Painting and Caulk???

We just gutted and are in the process of remodeling our entire bathroom. We caulked all the seams in the trim, crown molding and base molding. However, the caulk was NOT paintable! After taping and painting the bathroom walls RED (another big job) we discovered the paint did not stick and actually peeled back when we removed the tape. Can I just recaulk with paintable caulk on top of the last job or do I need to remove all remains of the first caulk and start completely over again with the paintable caulk before trying to paint the edges again? What a mess!

It sounds like you painted over OIL paint. That could be the reason that the paint did not stick. Light sand and a oil primer will help:)
I would remove the caulk before putting new stuff on as well.

It is possible that the paint peeled because it hadoil base underneath but often tape removal will pull paint with it. A bond is formed with paint from the wall to the tape. It is best to remove tape before the paint dries.

It is also best to remove the old caulk. Take an utility knife and cut both sides of the caulk, that should help to remove the majority of it.

You could also prime the caulk with pigmented shellac and then paint but removal is best.

It's best to remove the incorrect caulk and replace it with the proper kind

It's also better to not use tape
If you must tape, take a look at ProWallGuy's sticky at the top of the "Painting" page

If you had to paint the non-paintable caulk, I've had success with wiping it with alcohol and putting a coat or two of pigmented shellac on it
Then top coating with a coat or two of paint

Usually that's for old caulk (well, I don't usually use the wrong kind on fresh work lol ;))
As the caulk is fresh, it should come off