The sentence is independent- not a fragment- because it contains a subject and a verb, and it expresses a complete thought. It can be a noun (e.g. “Harry”) or a pronoun (e.g. “He”), singular or plural, and the verb can be past, present, or future tense.
But even though the sentence has a noun (or pronoun) and a verb (adjective or adverb), it must also express a complete thought. Saying “I am” meets the first criteria, but not also the second. Hope that helps!