I was upset more than 15 minutes. or I was upset for 15 minutes and a bit more. "добрые/добрых" - часто означает "как минимум/по крайней мере/at least", "добрых _smth_" = "full/all/complete _smth_ and a bit more" _smth_ - period of time/amount/quantity Напр.: Это весит добрый килограмм. It weighs more than one kilogram./ It weighs one kilogram and a bit more.

"I was upset for a well quarter of an hour." But I am afraid it sounds strange in Russian. I would rather say: Я был огорчён добрых четверть часа.

It's sounds strange,but it's not wrong!))