A lot of it has to be context. If I'm talking to someone new, I'm going to try very hard to read very little into the text they've sent me. If it's a message from my partner -- yeah, every aspect of that message can communicate something to me. And in that latter case, a period on a short statement is a warning sign equivalent to passive-aggressive "I'm fine."
Another thought is it literally changes how I read a sentence. Considering "I'm fine." vs "I'm fine" vs "I'm fine...", the first ends abruptly and is cut off. The second ends more gently and naturally. And the last trails off implying...something depending on the person and context. Consider poetry. No punctuation, a comma, a dash, a semi-colon, or a period each imply a type of pause (or lack of pause) at that moment in the words. And in poetry, that can mean everything.