You can re-encrypt any cipertext witout tampering the original encryption as long as you use independent keys for the two encryptions. A formal proof can be found in the paper
Cascade ciphers: The importance of being first by Massey and Maurer.
The paper also shows that a cascade may not be as strong as the second cipher. This is a surprising observation and it is difficult to come up with a case that is not totaly unpractical.
For example, if the first encryption uses compression then the length of the ciphertext could reveal some information. Assume for example that a bank is sending a standard letter to their customers with new pin codes. The letters where the new pin code overlaps e.g. with the banks zip code might compress better. Hence an attacker seeing a series of encrypted letters could potentially select the ones with easy guessable pin numbers.
In such a case, re-encrypting the letter a second time might not help.
Note, this does not mean a second encryption does never help. Just that there is no guarantee that the cascade is as strong as the second encryption.