Yes. In other words, any HTTP request message is allowed to contain a message body, and thus must parse messages with that in mind. Server semantics for GET, however, are restricted such that a body, if any, has no semantic meaning to the request. The requirements on parsing are separate from the requirements on method semantics.
So, yes, you can send a body with GET, and no, it is never useful to do so.
This is part of the layered design of HTTP/1.1 that will become clear again once the spec is partitioned (work in progress).
Yes, you can send a request body with GET but it should not have any meaning. If you give it meaning by parsing it on the server and changing your response based on its contents, then you are ignoring this recommendation in the HTTP/1.1 spec, section 4.3:
...if the request method does not include defined semantics for an entity-body, then the message-body SHOULD be ignored when handling the request.
And the description of the GET method in the HTTP/1.1 spec, section 9.3:
The GET method means retrieve whatever information ([...]) is identified by the Request-URI.
which states that the request-body is not part of the identification of the resource in a GET request, only the request URI.
The RFC2616 referenced as "HTTP/1.1 spec" is now obsolete. In 2014 it was replaced by RFCs 7230-7237. Quote "the message-body SHOULD be ignored when handling the request" has been deleted. It's now just "Request message framing is independent of method semantics, even if the method doesn't define any use for a message body" The 2nd quote "The GET method means retrieve whatever information ... is identified by the Request-URI" was deleted. - From a comment
From the HTTP 1.1 2014 Spec:
A payload within a GET request message has no defined semantics; sending a payload body on a GET request might cause some existing implementations to reject the request.