Although I am not (by any stretch) an expert on finite simple groups, let me flesh out my above comment.
Consider the following QCFSG (i.e., "qualitative" CFSG): with only finitely many exceptions, every finite simple group has prime order, is alternating, or is one of the finitely many known infinite families of Lie type. QCFSG must have been conjectured rather early on, whereas the exact statement of CFSG was much harder to come by, as much of the early work on the classification problem resulted in discovery of new sporadic groups.
I guess that early on someone must have looked at the nonsporadic finite simple groups and noticed that, except for the two exceptions listed above, they have distinct orders. [Assuming this is actually true, that is. I have no reason to doubt it, but I haven't checked it myself.] Once you notice that, if you believe QCFSG, then you certainly think that the order of a simple group determines the group up to finitely many exceptions. It is very hard for me to imagine how you could prove that the number of exceptions is precisely two without knowing the full CFSG.
I cannot resist conveying a story of Jim Milne, whose moral is that you shouldn't feel too bad when you say something absolutely stupid in public: better mathematicians than you or I have said stupider things.
Finally, a story to keep in mind the next time you ask a totally stupid question at a major lecture. During a Bourbaki seminar on the status of the classification problem for simple finite groups, the speaker mentioned that it was not known whether a simple group (the monster) existed of a certain order. "Could there be more than one simple group of that order?" asked Weil from the audience. "Yes, there could" replied the speaker. "Well, could there be infinitely many?" asked Weil.
For the source, and for some further fun stories, see