In reply to LastBoyScout:
I'm a painter and do plaster board repair, and in defense of your man there, it doesn't sound like his process is necessarily at fault. Patch the hole, add some mud, seal it, then paint it. It's pretty basic stuff. Of course I can't comment on how well he executed it, but you're obviously not happy with it. (Btw, that Zinsser Bin sealer is some of the best stain blocker you can use, so rest assured that bit is fine.)
Did you try and speak to him about it? If a customer of mine is not happy, or there is an unforeseen problem with the job (it happens), I'm back over there fixing it no questions asked, and for free if it was due to an oversight on my part (it happens). I would speak to him first and give him the chance to set it right.
Assuming you're beyond that, I'll try and help with some suggestions of what to do. You can definitely get it done in a couple days.
I can't perfectly imagine the situation, but it sounds like there is exposed paper tape, and there shouldn't be exposed paper tape. It will definitely peel and bubble if it gets wet, it's paper after all. The layers at the joints should be: plaster, tape, thin layer of plaster (sanded flush at the edges), PVA/primer, top coat. If you're putting top coat straight onto bare plaster/filler, or worse paper tape, it's not gonna work well or look good. Fill the surface first, sand it all flat, PVA/primer it, then paint over top.
When sanding, be careful not to sand all the way through the plaster down to the tape. Sounds like maybe that's what happened? If so, you need to get some joint compound, apply a thin layer back over the joint with your trowel, give yourself a good margin either side of the tape underneath so you can sand the edges flush w/o getting down to the tape. Let it dry 24 hrs at least, do the sanding, PVA, then paint.
I hope I've understood the issue correctly, and hopefully explained the process clearly.
Good luck!