No you can't. There are many threads about this like this one-http://www.chess.com/forum/view/help-support/bug-in-deleting-messages . Chess.com is working on it.
/Nice avatar
Can't delete messages anymore?
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The delete button is gone. An 'archive' button is its replacement. So, we can't delete general messages to us anymore, only archive them?
What prompted that change?
Thanks.