Consecutive hot dogs

Here is a puzzle I made up after hearing a nice number theory result from Steve Leth.

Puzzle

At the warm-up event for Oscar’s All Star Hot Dog Eating Contest, Alvin ate all the hot dogs he could manage. Bob then one-upped him by eating one more hot dog than Alvin ate. Not to be outdone, Carl ate one more than Bob. This continued with each contestant eating one more dog than the previous contestant, until in the end, a total of 253 hot dogs had been eaten. How many contestants participated in the event?

Extra credit

After hearing about the success of the event, Nathan claimed that he ran an almost identical event with a different group of contestants. Again, each contestant ate one more hot dog than the previous contestant, but upon completion, 256 hot dogs had been consumed. How do you know that Nathan is lying?

Extra extra credit

For which total numbers of hot dogs is it possible to determine the number of contestants participating in such an event? Assume there must be at least three contestants?