A good pair of shoes is the only barrier between you and chronic pain. Running involves a lot of strenuous impact that most modern shoes try to cushion with specially engineered foam. But foam, being foam, is prone to breaking down — usually after around 300-500 miles of use. After that, your shoes stop absorbing shock, which means they stop working the way they were designed.
The math might give you slightly more time, but most experts agree on replacing your running shoes every 6-8 months, which also accounts for breakdown in other parts of the shoe, and any localized wear depending on your gait pattern or your workout intensity.