The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding non-significant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.