We analyze American put options in a hyper-exponential jump-diffusion model. Our contribution is threefold. Firstly, by following a maturity randomization approach, we solve the partial integro-differential equation and obtain a tight lower bound for the American option price. Secondly, our method allows to disentangle the contributions of jumps and diffusion for the early exercise premium. Finally, using American-style options on the S&P 100 index from January 2007 until December 2012, we estimate various hyper-exponential specifications and investigate the implications for option pricing and jump-diffusion disentanglement. We find that jump risk accounts for a large part of the early exercise premium.