Can you make yourself understood in English using only three tenses? Probably.
You can probably make yourself understood using nothing but the present indicative and other words expressing past and future. You won't be speaking good English but you can probably make yourself understood.
Do English speakers use only three tenses in conversation? No.
This would be very natural and not at all advanced:
Wife: "When you were shopping at the supermarket, did you remember to get apples?"
Me: "If I'd seen any Galas that looked good, I would have gotten them, but the only decent ones were Granny Smiths and I know you don't like them. I might have had more luck at Whole Foods but they're always so expensive."
Wife: "Well, why don't you go to the farmer's market this afternoon? Maybe they'll have apples."
I'm not enough of a grammarian to know how many verb tenses I used in that example, but I think it must be at least six.