The paradox tells them that the perpetrators used a Time-Turner to send messages back in time. TTs are generally locked to a single person's use; someone else can't come along, steal or take the TT by force, and use it for themselves. They explicitly say they suspect someone might have forced Harry to use his TT in this way. They are fools not to check if someone forced or tricked another child to use their TT.
Also, they check if Harry can still use his TT that day because they suspect him. But they do not treat him as an adversary. Flitwick doesn't question him closely to make sure the Harry in front of him really just came back 6 hours in time. It would be trivial to ask him questions interactively to prove that, instead of just have him pass a message. Minerva should have had the message say: "Filius, please question Harry to make extremely sure that he just came back 6 hours in time!"
Indeed, Dumbledore and his allies are being somewhat less than maximally clever here... it would make more sense if they simply didn't know about the other student with a Time Turner because it was an unauthorized copy, or something. (And it's quite apparent that Time Turners are far more common in the MoR-verse than in canon, Hermione was only able to get one in book 3 was said to be a very unusual event which was only happened because the staff had an extraordinarily high level of trust in her.)
Update: Discussion has moved on to a new thread.
After 61 chapters of Harry Potter and the Methods of Rationality and 5 discussion threads with over 500 comments each, HPMOR discussion has graduated from the main page and moved into the Less Wrong discussion section (which seems like a more appropriate location). You can post all of your insights, speculation, and, well, discussion about Eliezer Yudkowsky's Harry Potter fanfic here.
Previous threads are available under the harry_potter tag on the main page (or: one, two, three, four, five); this and future threads will be found under the discussion section tag (since there is a separate tag system for the discussion section). See also the author page for (almost) all things HPMOR, and AdeleneDawner's Author's Notes archive for one thing that the author page is missing.
As a reminder, it's useful to indicate at the start of your comment which chapter you are commenting on. Time passes but your comment stays the same.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: