It it possible a cold sore could get infected?

Yes: A herpes virus infection (cold sore) can result in a superficial skin infection called impetigo. It is not common, but it can occur. If your cold sore is not healing as you expect from past experience, i would see your pcp or a dermatologist. Sometimes, but not often, the cold sore virus can persist and requires a different approach to treatment to resolve. Good luck to you.

Yes: While a cold sore is caused by the herpes virus, like any wound, it can become secondarily infected by bacteria. You could try several days applying Bacitracin ointment (not neosporin as it can irritate the skin) several times a day. If it doesn't heal during that time, you should get it looked at.