The multiple hypotheses testing problem is studied when the family of underlying probability distributions involves a nuisance parameter. A necessary and sufficient condition for the existence of an adaptive decision procedure is obtained when the nuisance parameter takes values in a compact topological space and observations (possibly in continuous time) have log-likelihood ratios obeying the large deviatons principle. By definition, an adaptive procedure is asymptotically fully efficient for each value of the nuisance parameter and does not depend on it. The mentioned adaptation condition is illustrated by Gaussian and Markov processes