Here's an excellent, albeit long, explanation of how best to mount a tire and tube by Lennard Zinn that appeared in Velo News.

Dear Lennard, Last season I experienced two sudden front wheel flats on long, two-mile, twisty and steep descents on hot summer days. Neither of these flats were puncture-related but instead were caused by the inner tube/tire blowing off the rim because of heat.
-Terry

Dear Terry,
I am willing to bet you that both of those flats were caused by the inner tube being caught under the rim bead. And catching the tube under the bead is far more likely to happen with a tight tire that you have to mount with levers. There is no way to get a long rip in a tube like that without blowing the tire off of the rim, and there is no way to get a tight tire to blow off of the rim without getting some inner tube underneath it to lift it off. Yes, the heat of braking precipitated the occurrence, but it was the trapped inner tube edge that, once the pressure came up enough due to the heat, blew the tire off.

When mounting any tire, but especially a tight tire, you need to make sure to finish at the valve stem, and not start at the valve like so many people do. This will significantly reduce the chance of that happening. Not only is tire installation best by finishing at the valve stem, but removal of the tire is also most easily accomplished by starting near the valve stem. That way, the beads of the deflated tire can fall into the dropped center (?valley?) of the rim on the opposite side of the wheel, making it effectively a smaller-circumference rim onto (or off of) which you are pushing the tire bead. If you instead push the tire bead onto (or off of) the rim on the side opposite the valve stem as a high percentage of people do, the circumference on which the bead is resting is larger, because the valve stem is forcing the beads to stay up on their seating ledges opposite where you are working.

With a standard, tubed clincher tire, there is another reason to finish at the valve stem besides ease of mounting, namely exactly what I think happened to you: ending up with a little bit of inner tube trapped under the tire bead where you finished pushing the tire onto the rim. You may even have taken the (appropriate and highly recommended) precaution of starting with some air in the tube to keep it from twisting and getting under the bead as you push the tire on, finally deflating it as is normally required only when it prevents you from getting the final few inches of the tire bead on. However, the edge of the flat tube can slip under the bead edge as it pops into the rim at that point, and this is even more likely to occur with a thin, latex tube.

When you pump it up to pressure with some tube caught under the bead, it may immediately blow the tire off of the rim (temporarily deafening you, the explosion is so loud), or it may explode at some point soon while riding. In either case, you will have an un-patchable tube, just like the two you?re now the proud owner of, with a long rip down its length. And, as you have now seen firsthand twice, you may experience risking your life if it is your front tire blowing at high speed, especially on a turn.

Minimize the likelihood of an explosion by finishing installing the tire bead at the valve stem from either side. The edge of the deflated tube may still slip under the tire bead as you push the last bit on, but it tends to do it less because the tire is less tight when mounted this way. But those caught tube edges, if they?re there at all, will be under just the final sections of bead extending in either direction from the valve stem. If you then push up on the valve stem after the tire is on, you can suck the adjacent sections of inner tube completely into the tire chamber and ensure that none is caught under the edge, waiting to blow the tire off of the rim. In any case, before pumping, always inspect around the edges of the tire bead by pushing inward on them so that you can see down into the rim alongside the tire.