We study the phenomenon of avoided crossings of eigenvalue curves for boundary value problems related to differential equations of Heun's class. The eigenvalues are given explicitly in asymptotic form taking into account power-type as well as exponentially small terms. It is exhibited that the phenomenon of avoided crossings of eigenvalue curves show a 'periodical' structure in the sense that at any integer value of the additional controlling parameter an infinite (in the sense of a large parameter) number of avoided crossings take place simultaneously. Some relations to other phenomena of the asymptotics of exponentially small terms are discussed at the end of the article.