It's impossible to NOT Communication
Common usage of language contains phrases like "He doesn't communicate with me", or "He's uncommunicative", giving us the impression that it's possible to NOT communicate. In fact, that's impossible. It's possible for someone not to talk to another person, but it's not possible to NOT communicate anything at all.
That's because communication does not involve just words, but it also is related to behavior, and unless one is dead, one always "behaves". Even staying still is a behavior. Silence communicates. Our bodies communicate non-verbally. So, so long as there is life there is communication, even if the person is intentionally trying to cease all communication.
In addition to sender side issues, keep in mind that human beings, and probably many living things are wired to extract meaning from the environment, so meaning will be created regardless of the intent of the sender. The receiver is wired to receive communication, however subtle.
Communication As Meaning
If you think of communication as meaning, things make more sense, since meaning comes from the "sender's" behavior, and the "receiver's" behavior. Even if the sender isn't actively transmitting (let's say sitting still without any obvious activity at all), the RECEIVER will attempt to interpret and create meaning from that.
The process of interpreting never stops, and while EFFECTIVE communication may not be occurring, communication is going on.
So, the answer is no. One cannot NOT communicate.