"Any PCIe device can run down to just a x1 connection as long as it is the first x1 lane, PETp/n0 and PERp/n0 (this is called ""Down-plugging""). The problem you're going to run into is the x16 card won't fit into the slot unless it is a x16 slot. If you're comfortable with modifying your motherboard you can cut out the end of the PCIe slot so that the card fits in (but don't cut the card itself) or get a x1 (or x4/x8) to x16 adapter board (which won't fit inside a chassis). There is also potential clearance issues with the card on the end that sticks out off the shorter slot. Also, a lesser concern is a high power card (75W) plugged into a 25W slot.
Below is an open ended x4 PCIe slot (in the middle closest to the ""ASRock"" logo). Notice that there are no electrolytic capacitor cans in the way if a x16 card were to be plugged in."