Yes, there are exceptions.

If you read Federal Rules of Evidence 803 and 804  you will see a list of exceptions. Many states have identical exceptions.

By way of example, consider the “excited utterance.”  This is an excited statement made in response to a startling event or condition.

For example, let’s say immediately after being hit by a rock, Gary yells, “Mark threw a rock at me!”

A startling event.

A startling event.

An excited utterance.

An excited utterance.

Gary’s statement that Mark threw a rock at him can likely be admitted into evidence to prove the truth of the matter asserted.  That is, the statement is probably admissible to prove that Mark threw the rock.  

The statement would be admissible as an exception to the hearsay rule because it is an excited utterance.