The possibility of exciting laser modes such as Laguerre-Gaussian (LG) or Ince-Gaussian (IG) modes is discussed on the basis of a gain-matching integral. We reach the conclusion that, using tight pumping and away from degeneracy regions, only the IGen,n modes can be excited. Furthermore, pure high-order modes with circular or elliptical nodal lines can never be excited. Only an approximation of such modes, which we call quasi-IG or quasi-LG modes, can be observed and only when the cavity is partially degenerate. We provide experimental results in perfect agreement with the theory and discuss the exact nature of the profiles observed at degeneracy in our experiments and elsewhere in the literature.