[This is what Wolfram tech support told me:]

That is because using machine precision (inexact) numbers such 0. (as opposed to exact numbers like 0) forces Eigenvectors[] to seach for eigenvectors numerically and hence the error message. 
For example, please evaluate and compare the results of the following expression used with machine precision and exact numbers:

    Eigenvectors[{{Cos[1. x], Exp[I x]}, {1, 0}}]

    Eigenvectors[{{Cos[1 x], Exp[I x]}, {1, 0}}]