Multichannel dynamical amplitudes can have CDD poles when viewed as the solution of the appropriate one-channel inelastic problem. Manifestations of this phenomenon are investigated. The technique is to work via the eigenamplitudes, in terms of which the essential many-channel dynamics take a simple form; for them, individual Levinson's theorems are derived. On applying the method, it is found that, for a wide class of cases, the phenomenon always occurs in at least one channel and, in many cases, further analysis enables one to say in which one. A speculative application is made to SU3 bootstraps.