No. To cause a nuclear explosion you need a chain reaction. A chain reaction requires a high enough density of fissionable material so that any one reaction is statistically likely to cause more than one additional reaction.
Unfortunately, for most nuclear materials, that is virtually impossible. It's almost as though nature doesn't want it to happen. Very few materials can undergo a chain reaction in the first place. Those that can, like Uranium, typically needs to be intentionally enriched to an extra high density of fissionable atoms before it can be used in a bomb. Worse, as you bring two pieces of fissionable material together to create a critical mass, the energy caused as they come together has a tendency to force the two pieces apart (such as by blowing them apart with vaporized material). If this occurs, only a tremendously small portion of the fissionable material actually does anything like an explosion. Most of it just heats up a whole lot.
In nuclear bombs, we not only have a lot of explosives to push the fissionable material together to create a critical mass, it is very carefully shaped explosives with extraordinarily tight initiation timings that makes sure a large portion of the mass undergoes a chain reaction. The odds of this happening are so slim that it would be simpler to simply round down and call it "none."
The best hope I think you'd have would be to use the heat of the nuclear pile to superheat a small pocket of water and have a steam explosion. Mythbusters did it to a water heater once or twice. It's no nuclear explosion, but it's at least good for TV ratings!
EDIT: Reviewing additional content related to this answer, I came across a fascinating Wikipedia page on Prompt Criticality. In any nuclear system, "supercritical mass" is defined to be the mass at which point the nuclear material will undergo a chain reaction. It is dependent on many things such as density, shape, and even nearby neutron reflectors. What I just learned from the page on Prompt Criticality is that there are two definitions of supercritical: prompt-criticality and delayed-criticality. Prompt-criticality is what matters for bombs. This is when the "prompt-neutrons" emitted directly from the fission event are likely enough to cause another event that a chain reaction occurs within milliseconds. For nuclear reactors, and your nuclear waste pile, delayed-criticality is more of an issue. Many of the remains of uranium fission are, themselves, unstable, and emit many more neutrons over the course of minutes. For a nuclear reactor, or nuclear waste pile, if the pile is delayed-critical, it will eventually melt down if you don't do something about it. However, because it isn't prompt-critical, it will not cause a nuclear explosion.
Who would have thought nuclear science was this complicated!