Even the cheapest NICs can be shown to do more than 750 Mb/s in synthetic tests with some tuning effort, which is more than what typical Windows file transfers do even with the highest-end server NICs. From this point of view, there isn't a strong argument for NIC swapping. The typical case is much worse, and the bottleneck is generally not the NIC or other parts of the network, but the OS, drive subsystem, and other hardware / software.
It's possible for some NICs to simply perform better than others in adverse conditions such as bad cabling and it's possible to mess up anything, so it's possible to have untuned drivers, bad NIC settings (e.g. duplex mismatches), or even bad connections, which can appear to magically improve when you drop in a replacement with new drivers and default settings.
It's also possible for some older hardware/software platforms to be so constrained that even a 10 GbE NIC (if a compatible one could be found) would give virtually no performance improvement.