So I decided to upgrade my GPU from an HD7870 to a MSI 6GB GTX1060 (short card version) because the HD7870 couldn't really handle shadows. The swap seemed to go well and performance was about twice as fast. But I noticed something odd: if I stressed the card with a zoomed view into lots of trees the frame rate would start OK but slowly, over many seconds, decline to even below what the HD7870 could do. I opened the case to check if the fan was not turning, but I couldn't find the fan! Then I noticed the fan was covered with a sticker that said Play Hard, Stay Silent and in small lettering "Remove before Gaming".
Worked much better with the fan uncovered.
I'm such a dummy. (Although, what on Earth is the point behind covering the fan with a sticker when the card is shipped in a sealed bag?)