The usual assumption that the specific heat of a paramagnetic crystal can be written as the sum of a lattice specific heat and a spin specific heat needs modification when there is strong spin-lattice coupling. The necessary theory is developed for two simple examples, and it is shown to lead to an increase in the maximum of the Schottky specific heat. It is suggested that the previously unexplained magnitudes of the specific heat maxima in cerium and praseodymium ethyl sulphates arise from strong spin-lattice couplings.