In PoA, Hermione mentions that herself and Harry must take special care not to let their past selves see them, because wizards have died form going back in time and being killed by themselves. However we see that everything Harry and Hermione do when in the past, has already happened. Like a self-fulfilling prophecy. But theres also the Grandfather paradox to worry about; If you go back in time and kill your grandfather, you would never had been born so you wouldn't be alive to go back in time and kill him the first place.
I tried my best to create a scenario that explains my theory on how it could be possible. If you can find anything wrong with it, please point it out in a review. But whether it really makes sense or not, I give you, for your reading pleasure:
The Tragedy of Time-Turner Joe
May 1st: 7:53am. Joe is walking down a seldom trodden path in the middle of nowhere when a man stumbles out of the bushes in front of him looking frazzled. Joe asks the man if he is okay and grabs his arm to help steady him. Shocked, Joe finds he is looking at a replica of himself!
The man looks at Joe, mouth gapping, and starts screaming illegibility and pulling away in horror. Joe in his surprise and fear, suspecting Dark Magic, a doppelganger, or someone sneaking around in his form c/o Polyjuice Potion, Joe strangles the man and hides the body in a near by ditch.
Being completely freaked out, Joe runs home.
May 19th: The events of earlier in the month virtually forgotten, Joe goes to work and finds out he is in danger of losing his job if he doesn't catch up on the work he'd procrastinated in favor of taking long walks in the middle of nowhere and picking off strange men who look like him.
Instead of getting right on it, Joe thinks 'Hey! Why should I exhaust myself by doing this all in one or two days? We're wizards! We have magic to take care of stupid problems like this!
After a day of lounging around, flirting with every female in the building, and general goofing off, Joe stops by the Ministry, fills out the proper forums, takes the eye, written, and practical tests and goes home with his newly rented Time-Turner.
May 20th: 8:00am. Joe sneaks off unseen to a small shack in the middle of nowhere, where he knows neither he nor anyone else has been for years. He is surprised to find a large pile of nutshells and chewed paper piled around the room. Uncaring of those little details, Joe turns his Time-Turner sending himself back to—
April 15th: —taking special note of the time of his departure. As long as he doesn't show his face around civilization till 8:01am of May 20, he'll be perfectly fine.
April 19th: Joe, who despite being a lazy bum, is actually quite good at planning. He has managed a bit of his work already. Deciding to pace himself (he DOES have all the time in the world) Joe spends some time sitting out side his shack, breathing the fresh air, and feeding the cute little squirrel bits of his sandwich.
April 27th: Joe has managed a good portion of his work so far. At this rate he'll have time to spend just sitting and doing nothing (like he did the first time around). Joe decides this was officially the best idea ever. The squirrel is even letting him pet it now!
April 30th: Deciding to take the night off, Joe goes out for a walk. He wanders out past a little stream, deeper into the woods then originally planed. It's late at night when Joe realizes he has no clue where he is. In stead of wondering around aimlessly, Joe decides to bunk down for the night in the middle of the woods.
May 1st: 7:53am. Groggy from sleep, Joe wakes with the utmost desire to heed nature's call. He stumbles out of the woods onto a seldom trodden path in the middle of nowhere.
May 23rd: Joe hasn't shown up for work in three days. One of his coworkers, Nancy, gets worried and decides to make sure he is alright. There is no one at Joe's home so; Nancy asks Joe's neighbors if they know where she might find him. One of the neighbors remembers Joe owned an old hunting shack out in the middle of nowhere, and perhaps Nancy might find him there. Nancy thanks the neighbor and apparates to the closest point she can.
After wandering around, Nancy is completely horrified to find Joe's dead body dieing rotting in a ditch. She quickly alerts the Aurors.
May 25th: The Medical Examiner finds that Joe's body has decayed what would be about the equivalent of a full month, as apposed to the three days since he was last seen. The Medical Examiner isn't concerned, however, as the small hourglass necklace around Joe's neck, and the affirmative comparison of the marks on Joe's neck to his own hand, make it quite clear what has happened.
Joe has been the victim of Accidental Time Travel Related Suicide.