We study unboundedness properties of functions belonging to generalised Morrey spaces Mφ,p(Rd) and generalised Besov-Morrey spaces Nsφ,p,q(Rd) by means of growth envelopes. For the generalised Morrey spaces we arrive at the same three possible cases as for classical Morrey spaces Mu,p(Rd), i.e., boundedness, the Lp-behaviour or the proper Morrey behaviour for p<u, but now those cases are characterised in terms of the limit of φ(t) and t−d/pφ(t) as t→0+ and t→∞, respectively. For the generalised Besov-Morrey spaces the limit of t−d/pφ(t) as t→0+ also plays a rôle and, once more, we are able to extend to this generalised spaces the known results for classical Besov-Morrey spaces, although some cases are not completely solved. In this context we can completely characterise the situation when Nsφ,p,q(Rd) consists of essentially bounded functions only, and when it contains regular distributions only.