Yes, you can put a PCIe x1 card in a PCIe x16 slot.
The PCI Express link is built around pairs of serial (1-bit), unidirectional point-to-point connections known as "lanes". This is in sharp contrast to the PCI standard which is a bus-based system in which all the devices share the same bidirectional, 32-bit (or 64-bit), parallel signal path.
The number of lanes actually connected to a slot may also be less than the number supported by the physical slot size. An example is a x8 slot that actually only runs at x1; these slots will allow any x1, x2, x4 or x8 card to be used, though only running at the x1 speed.