I'm feeling mischievous.  The history of the Kervaire invariant problem is strewn with false proofs, Jim Milgram's is only one of many.  A student of mine (who I will leave nameless) had a preprint (around 1980?) that solved the problem but that Mark Mahowald quickly shot down.   The most recent example I know of is that of a Russian mathematician (who I will also leave nameless).  See Math Reviews MR2590025 (2010k:55031) 
Differentials of the Adams spectral sequence and the Kervaire invariant  (Russian) Dokl. Akad. Nauk 427 (2009), no. 5, 601–604; translation in Dokl. Math. 80 (2009), no. 1, 573–576.  From the text (translated from the Russian): "In this paper, we study the differentials of the Adams spectral sequence for stable homotopy groups of spheres and solve the Kervaire invariant one problem for n-dimensional manifolds when $n=2^i-2$, $i\geq 6$.  That is a four page paper.  Would that it were so simple!