In this paper, we aim to obtain Massera type theorems for both linear and nonlinear dynamic equations by using a generalized periodicity notion, namely (T, λ)-periodicity, on time scales. To achieve this task, first we define a new boundedness concept so-called λ-boundedness, and then we establish a linkage between the existence of λ-bounded solutions and (T, λ)-periodic solutions of dynamic equations in both linear and nonlinear cases. In our analysis, we assume that the time scale T is periodic in shifts δ ± which does not need to be translation invariant. Thus, outcomes of this work are valid for a large class of time-domains not restricted to T = R or T = Z.