Tile linear abstract evolution equation (*) u'(t) = Au(t) + (t), t R, is considered, where A: D(A) E -> E is the generator of a strongly continuous semigroup of operators in the Banach space E. Starting from analogs of Kadets' and Loomis' Theorems for vector valued almost periodic Functions, we show that if σ(A) iR is countable and : R -> E is [asymptotically] almost periodic, then every bounded and uniformly continuous solution u to (*) is [asymptotically] almost periodic, provided e - λ t u(t) has uniformly convergent means for all λ σ(A) iR. Related results on Eberlein-weakly asymptotically almost periodic, periodic, asymptotically periodic and C 0 -solutions of (*), as well as on the discrete case of solutions of difference equations are included.