if x is an eigenvector for the matrix A, then Ax = λ1x, for some scalar λ1.
if x also satisfies Ax = λ2x, then we have: 0 = Ax - Ax = λ1x - λ2x = (λ1-λ2)x.
now eigenvectors cannot be 0 (by definition), so (λ1-λ2)x = 0 implies λ1-λ2 = 0: that is, λ1 = λ2.
so an eigenvector can only have ONE eigenvalue it belongs to.
it is possible, however, for an eigenvalue to have TWO (or more, linearly independent) eigenvectors.
consider the matrix I =
it should be clear that both (1,0) and (0,1) are eigenvectors with the same eigenvalue, 1, and that these two vectors are linearly independent.