In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational α > 0 we can construct a smooth g ∈ L 2 (R) such that for any two rationals a > 0 and b > 0 the collection (g na,mb ) n,m∈Z of time-frequency translates of g has a finite frame upper bound, while for any β > 0 and any rational c > 0 the collection (g ncα,mβ ) n,m∈Z has no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g ∈ L 2 (R) which is sufficiently well behaved, there exist a c > 0, b c > 0 such that (g na,mb ) n,m∈Z is a frame whenever 0 < a < a c , 0 < b < b c . We present two examples of a nonzero g ∈ L 2 (R), bounded and supported by (0, 1), for which such numbers a c , b c do not exist. In the first one of these examples, the frame bound equals 0 for all a > 0, b > 0, b < 1. In the second example, the frame lower bound equals 0 for all a of the form l · 3 −k with l, k ∈ N and all b, 0 < b < 1, while the frame lower bound is at least 1 for all a of the form (2m) −1 with m ∈ N and all b, 0 < b < 1.
