RE: Oh boy, this could be bad...
It could be several things.
Do they have steel cases? Steel is not nearly as elastic as brass and in a high pressure round like the 30-06 it may not have deexpanded back to its original shape.
A dirty chamber could lead to stuck case.
A improperly (over) loaded cartridge could certainly lead to a stuck case.
If the chamber is not polished properly (600 grit) it could be too rough and not let go of the case properly.
The cases could be out of SAAMI specs. If the cartridge is too short you headspace would be excessive for that particular cartridge and could cause it to rupture.
With out actually seeing it we all would only be guessing.
If you don't need to shoot it immediately I would let the gunsmith get it out. If you do need it right away get a COATED steel rod and lightly tap it out after puting some kind of lubricant down the bore and letting it work for a while.