The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.

The fact that the [Steiner-Lehmus Theorem][1] is false over the complexes shows that it cannot be proved using what <a href="http://cs.nyu.edu/pipermail/fom/2004-August/008394.html">John Conway calls "equality-chasing" arguments</a>.

This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.


  [1]: http://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem