Sometimes. Depends on the NIC.
And on the switch side the answer is generally yes, but the switch asic must support the lower speed.
Example of it not working... Copper GLC-T= into a N9372PX, so it should support 10/100/1000 but since the switch asic only supports 1000/10G/40G it won't allow the SFP to run at 100 speeds (sometimes you have OBM devices still at 100)
SFP is on the supported list for switch just not at all speeds.
On the card side I personally have never tried anything but the intended SFP modules for the card... A 10G SFP though will often run at 1G but not all. I found some IBM equipment where it's 10G only. (Again I assume that's all the asic support)
I will be getting some Intel (HP branded) cards shortly, I may true a few different SFP's see what works but most SFP I have sitting around are all Cosco branded so not sure if I really have much luck.