Can a 240V item trip the main breaker and not trip it's own breaker? I've got a 240V industrial fan connected to a separate breaker in my 100A panel.

if both breakers are in good condition then it should not happen. if one or both is not then yes, anything is possible...

If the 240V item is the cause of the main tripping (let's say, due to a short circuit in the fan), I agree with hfcarson. But how do you know that the load of the fan was not just the straw that broke the camel's bank? For example, if all your other loads operating and drawing current before operating the fan was let's say 85 amps, and you turned on the fan and it was using 35 amps (total 120 amp load on 100 amp main) the main could trip even though no individual breaker capacity was exceeded. Maybe I am missing something- help me out.

Just to level set your question.

The voltage required for the fan is basically irrelevant. The issue is the amperage needed (aka "load") needed.

But there are other considerations which may have to be considered, particularly the "start-up or in-rush" amperage needed to get the fan's motor running.

By the way, what is the amperage required by the fan?