Well, yes, at some point your body will give up. Assuming you don't run to the point of exhaustion, pushing to hard usually shows up later in the day, or the next day, as DOMS (delayed onset muscle soreness). Sometimes it will appear as actual injury: pulled muscles, strained tendons, aching joints, etc. Increasing mileage or pace (but not both at the same time) slowly and progressively will help to get you to your half-marathon without any issues. Increasing too fast can lead to injury and a long layoff. Hopefully you have a good mileage base and a good training plan to guide you to your goal.
Yes, overall fatigue and burn out and injury like Lenzlaw said are some of the main things that could happen. What I have heard and try to follow is the 10% rule of not adding more than 10% of the total amount of running to your next week. Allowing your body to get comfortable with the new mileage before adding more.
Also if you push too hard over a period of time, you may start to show signs of overtraining. These can be subtle at first but could include a higher than usual resting heart rate, unusual fatigue that has no other explanation, and coming down with annoying minor illnesses (colds, etc.). Listen to your body to avoid them.
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Angels Baseball Foundation 5K, Anaheim, CA, 24:24
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Have you thought about maybe picking up a training plan for 13.1?
Hopefully you are doing some easy run days, some harder run days and one long slow distance day (that should increase a tiny bit every week up to your 13.1). But I really like Lenzlaw's overall answer. I'd say just don't over do it or you'll also burn yourself out prior to race day.
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