We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q-quasisuperminimizers. More precisely, if ui is a Qi-quasisuperminimizer, i=1,2, where 1<Q1≤Q2, then u=min{u1,u2} is a Q-quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q. We provide the first examples of this phenomenon, i.e. that Q>Q2.In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q=Q1Q2 turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers.