Critical amplitudes A+ associated with the temperature variation of the heat capacity are analyzed by means of renormalization-group techniques in both position and momentum spaces. We describe a mechanism according to which the amplitudes A diverge as the critical exponent a approaches a nonpositive integer. In between two consecutive divergences at least one amplitude vanishes at least once. The coefficient P in the expansion A+ /A- =1—Pa+0 (a~) is computed by means of e expansion and Migdal-Kadanoff renormalization-group technique. Systems for which the critical exponent alpha is negative but larger than —1 exhibit either a cusped heat capacity if A+/A- >0 or a smooth maximum in the heat capacity at a temperature other than the critical one and an infinite slope at Tc, if A+ /A <0. Implications of this observation for the interpretation of experiments on random-bond systems such as Fel „Zn, F2 are discussed.