Yes, it is possible to take too much thyroid medication. When doing so you can start to feel hyper. Most people will feel jittery, fast heart rate, too much energy, insomnia, and more.
That is why it is extremely important to make sure you thyroid lab results are being compared to optimal thyroid lab ranges, not the lab ranges. From there you should be getting your medication dosage. If you are going by the lab ranges there is a chance you are being prescribed the wrong dosage which can lead to symptoms.