I have a beef with that pun... as the groan grows. It's real PUNishment.
For me, the sort of Humor that seems to be funniest when it has a component that is truth, or at least like enough to truth.
Usually when it exposes a new way to perceive something in an absurd way.
Generally, if you know it's false, or have had that perception already, the joke falls flat.
If it's some new way of seeing something that makes that absurd connection, it seems to become humor.
Something that has been loaded with absurd social pressure or emotion is usually the "funniest", for having a lot of different, but "forbidden" ways to think about it. It's why religion tends to be funny.
Now the following jokes may or may not be funny, yet, given the tragedy+time figure. These are examples of tragedy+time for most, but some found them funny shortly after the events. Forbidden, and emotionally loaded. They're 9/11/WTC based, so don't go if you think you'll be offended... because you almost certainly will be.
http://www.temple.edu/isllc/newfolk/bigapple/bigappleappa.html
Challenger/Columbia jokes...
http://www.netfunny.com/rhf/jokes/88q3/16840.12.html
The really 'awful' thing is, people are more inclined to forgive a horrible disaster joke than a cheesy pun... what's with that Mr. Peabody?