Of course, if a message is encrypted and on a secure and secret storage it will impossible to compromise its security, because if the message is not available at all to an adversary, how could he attack it?
By the way, if it is correctly encrypted with a key of reasonably length its security is not compromised even if the encrypted message is made available to the adversary. The confidentiality protection is one, the main probably, of the security property of the cryptography and the threat model is exactly this one. Usually in proving encryption schemes the property researcher look for is the semantic security, meaning that even a bit of information cannot be recovered from the ciphertext.
In pratice it depends on your encryption scheme. If the message is encrypted with something that is at the state of the art, well implemented there is no risk and you can sleep well with no worries.
If you want to estimate the time window of security you could check the KeyLength page.
For the same reason keeping several encrypted message togheter does not leak information and does not represent a risk. In security proofs an adversary is supposed to have access to a decryption oracle and he is allowed to ask for a reasonable number of decryptions (but the challenge message) and the encryption is still secure. On the other hand, if your public key is known by the attacker (and it is supposed to be known, as it is public) the attacker is able to encrypt whatever he wants with your key, so he is able to get a bunch of encrypted messages (even better, knwon plaintext and ciphertext).