Tire popping out of rim when inflating tube This is very similar to this question.

When I fill my tubes up to about 85 psi the tire pops off the rim and the tub gets caught on the rim and blows up.  I have gone through 4 tubes in two days at this point and the only thing that seems to help is to inflate them to about 60 psi.  Even putting pressure on the tires so they seat correctly (not sure if that is the correct term, I am quite new to cycling) only works until I put weight on the bike, and then the tubes blow.

For some info, I frankensteined a bike together out of used/old parts so my rims are quite old (although relatively true after ~8 hours of work) but the tires are falling apart.  The rims are 27's (inches?) and there were really no options for newer ones at the place I put my bike together at, only 700's.

Could replacing the tires fix this?  I would like to get the tubes to around 90 psi since at 60 they are quite flat when riding.

Edit:  Some details on the tire, tube and rim.

Rim:  ~1 Inch wide at the top, ~25 1/2 inches in diameter
I couldn't really figure out how to check the bead.
Rim tape looks fine, it is some hardish plastic covering the spoke screws.

Tire:  See these images of the tire.
Supposedly a 27 x 1 1/8 tire that holds 90psi.  Not sure what else is helpful info other than the pictures of the deteriorating side walls (and the lip at the bottom that catches the bead)

Tube:  700x18-23 27x1 48 mm presta valve.  I can't find max pressure, but I am sure it is in excess of 90 psi.
Three things to check:

1, Is the bead seated correctly. It's pretty common to get tyres blowing off the rim if they're not seated correctly. That said, this usually leads to an exploding tube rather than the tyre just coming off
2, As Daniel R Hicks said, are the rims so worn they're expanding?
3, Check the rim for dings and warping as Malarky sort of says, but not just the area for the bead. You may have a bent rim

Chances are, though, that the tyres are done. The pictures would make me think they're on their last legs. I had a pair of 20+ year old ones recently that ended up in the bin as they just weren't reliable enough for me. Some things aren't worth saving a few euros/pounds/dollars over, and the contact points on a bike are some of those things for me. If the rims seem fine new tyres are where I'd go.