According to my research, it is indeed possible to overdose on some amino acids. For instance, too much L-Isoleucine can cause hypoglycemia.
Here are some links to sites that provide more information:
http://health.yahoo.com/drug/d03862a1 (specifically about Magnesium amino acids chelate)
http://www.vitamins-supplements.org/amino-acids/ (click on the name of any of the listed amino acids for more detailed information, including signs of an overdose)
I hope this information is helpful. Please let me know if you need anything else. Thanks for posting your question!