No, it is not possible. Any modern encryption algorithm is designed so that even even if you have some ciphertext and part of the corresponding plaintext, it is unfeasible to find any information (other than the length) about the part of the plaintext that you don't know. This is true even if you get to choose all of the plaintext except one bit, and you get the corresponding ciphertext and you want to find the bit that you don't know. This property is called indistinguishability under chosen-plaintext attack (IND-CPA for short).
Unfeasible is a technical term, meaning that it requires an amount of computation that cannot be performed in practice. The theoretical definition is that the complexity of the attack is more than polynomial over the size of the key. In practice, key sizes are chosen so that it would take more than the total computational power available in the world for more than a lifetime to break the key (for example, it would take a billion PCs more than the age of the universe to break a 128-bit key).
Note that I simplified the definitions a bit; the actual definitions are fairly complex and subtle. If you want to understand them precisely, read a cryptography textbook.
In practice, this assumes that:
- There is no flaw in the protocol, such as using the same key for different purposes or accepting invalid output, that could lead to the implementation acting as an oracle.
- There is no flaw in the implementation, that could lead to secret information leaking directly or indirectly.
- The algorithm genuinely has the IND-CPA property. It is in fact not known whether any encryption algorithm (apart from the very unwieldy one-time pad) has this property. In practice, people use algorithms that cryptographers have spent many years trying very hard to break, and failed.
In practice, most security flaws related to cryptography are due to implementation bugs. A minority are due to protocol weaknesses, and modern protocols tend to be designed with more attention than protocols designed 20 years ago. Flaws in modern encryption algorithms are extremely rare.