Wikipedia gives an answer although it provides no reference.
The Bazeries cylinder was a relatively strong system at the time (compared to many other systems in use), and Etienne Bazeries, a competent but very opinionated man, is said to have regarded it as indecipherable. In fact, it is hardly impregnable, and the "Pers z" code-breaking group of the German Foreign Office cracked the M-138-A in 1944. However, by that time the Americans had much more sophisticated cipher systems in operation. Many of the decrypts of Allied communications presented to the German high command were disregarded, because all they did was confirm bad news that nobody wanted to acknowledge in an environment where defeatism was potentially a capital crime.
de Viaris (aka Marquis Gaetan Henri Leon Viarizio di Lesegno) who is famous for one of the first printing cipher devices (1874), solved the Bazeries cylinder in 1893, so Bazeries alleged confidence in the system was ill-placed.
One major weakness of the Bazeries cylinder is that the offset from the plaintext letter to the ciphertext letter for the cipher alphabet on each disk will be exactly the same.
It intuitively seems easy to brute-force with a partially known plaintext. The chance that two disks will have the same distance from $C_i$ to $P_i$ is negligible. You then brute-force the possible outcomes and compare them against a dictionary. You've arrived at a small subset of possible plaintexts so you select the one that makes more sense. A second message coded with the same key will verify the results.