The authors present a two-channel bag model of heavy quarkonia including their decay into mesons containing one light and one heavy quark. In the Born-Oppenheimer approximation they find that, if there is a massless quark-antiquark pair present in the bag, the system prefers to fission into two bags beyond a critical separation of the heavy quark-antiquark pair. The flavour thresholds of the psi - and Upsilon -resonances are quite well reproduced.