"Can I run?" That's gotta be the most-asked question a healthcare professional gets from a runner when they're injured.
My PT has taught me the concept of Active Rest. "Rest" is relative. Maybe you cut back on your running but still run some. Maybe you quit running for a bit but do something else. Go to the pool. Get out your bike. Go for a walk instead of a run. Or maybe you cut your weekly mileage in half. That's resting. And your fitness level won't deteriorate nearly as fast as you think it will.
Resting is a balancing act. The amount of rest needed is determined by what you're recovering from. You want to ease off to give your body a break and chance to heal. If you listen closely enough you'll know if you're resting appropriately. Interestingly, however, movement promotes healing. So find that level and type of activity that will promote healing but not put you in a deeper hole.