Let S be a finite generalized quadrangle (GQ) of order (s, t), s<>1<>t. A k-arc K is a set of k mutually non-collinear points. For any k-arc of S we have k=<st+1; if k=st+1, then K is an ovoid of S. A k-arc is complete if it is not contained in a k'-arc with k'>k. In S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984, it is proved that an (st-m)-arc, where -1=<m<t/s, is always contained in a uniquely defined ovoid, hence it is a natural question to ask whether or not complete (st-t/s)-arcs exist. In this note, we prove that the classical GQ H(4, q 2 ) has no complete (q 5 -q)-arcs. We also show that a GQ S of order s with a regular point has no complete (s 2 -1)-arcs, except when s=2, i.e. S Q(4, 2), and in that case there is a unique example. As a by-product there follows that no known GQ of even order s with s>2 can have complete (s 2 -1)-arcs. Also, we prove that a GQ of order (s, s 2 ), s<>1, cannot have complete (s 3 -s)-arcs unless s=2, i.e., S Q(5, 2), in which case there is a unique example (up to isomorphism).