Can you get your heart rate too high by running?

Depends. If you have coronary artery disease, you could cause symptoms. If you have an arrhythmia you could have a extremely heart rate. However, usually simply exercising or running will not give you a heart rate that is too fast.
Yes. Yes you can, but not a common thing to get your heart rate abnormally high.

Related Questions

Why is my heart rate so high while running? Could I be in danger of a serious arrhythmia?

Objective values. Some people are more heart aware than others. Sometimes patients think they have a high heart rate when in reality it is within normal and appropriate range. Objective evaluation by pulse meter or electrocardiographic monitoring is necessary to fully answer your question. Ask your health care provider to evaluate you at rest and during exercise to relieve your concerns. Read more...

I just started working out and while running on the treadmill my heart rate went up to 180 is that considered too high?

Maybe. The maximum heart rate is calculated by subtracting your age from 220. For a 33 year old this would be 187. For vigorous physical activity it is recommended to exercise at 70-85% of the maximum heart rate. For a 33 year old that would be about 131-159. For your age 180 is not above the maximum recommended but above the recommended range. Read more...