Put the cartridge in firmly but don't force it down. Push the cartridge to the leftmost position once in the slot, then tilt up the right side of the cartridge ever so slightly. If it doesn't work, trying pushing the cart backwards just a little, or even wedge paper or cardboard to help it stay leaning back. Last resort is to gently pull the cartridge back up out of the slot, but this by a hair each time.
This works for SNES and N64. Down, left, up-right, back, and up. All very gently.