Shower thought about density of states

In my old quantum class we used the WKB approximation to calculate the time it would take for a PBR can to tip over due to thermal fluctuations. The tunneling probability is exponentially suppressed and goes (roughly) as e^{-barrier width*height}. We of course got an extremely small number.

In the calculation, though, we assume that there is only one way for the can to tip over. If we were being honest, we should've integrated over a density of states up to infinite energy to account for all possible ways it could tip. This of course wouldn't make much of a difference because DOS's are usually polynomial or sub-polynomial.

However, if the DOS grew at a faster rate than the energetic suppression, then the beer's time-to-tip would go to zero. My question is if there are any systems where the DOS blows up, enabling us to see macroscopic quantum effects, and, if not, is there any fundamental reason why a system's DOS can't blow up beyond the observation we don't see weird stuff everyday? Does this line of thought place any useful bounds on new physics?