We investigate lower bounds on the subgeometric convergence of adaptive Markov chain Monte Carlo under any adaptation strategy. In particular, we prove general lower bounds in total variation and on the weak convergence rate under general adaptation plans. If the adaptation diminishes sufficiently fast, we also develop comparable convergence rate upper bounds that are capable of approximately matching the convergence rate in the subgeometric lower bound. These results provide insight into the optimal design of adaptation strategies and also limitations on the convergence behavior of adaptive Markov chain Monte Carlo. Applications to an adaptive unadjusted Langevin algorithm as well as adaptive Metropolis-Hastings with independent proposals and random-walk proposals are explored.