The only time I see real problems with the cheapo on-board NICS anymore is on systems that have had their clock speeds played with a lot (like in overclocking), since they are often tied directly to system clock values. For the price you pay for them, they work great.
The problem with NIC testing is that you are contrained in speed and reliability by the worst device on the network. For most internet users, the bottleneck is usually the crappy gateway/multi-port router/switches ISP's sell you. For internal-only LAN connectivity, it is usually the cheapo hub/switch (often the same crappy gateway/multi-point router/switch) between the PC's.
Even if you have great in-between devices, for NIC's to test/work best, you really need the exact same NICs or onboard chips on each end of the connection points to make true apples to apples comparisons. Having a kick-ass NIC on one machine connecting to a cheapo NIC on another machine doesn't tell you much about either NIC's true performance. Even better when "testing" NIC's is a using a straight cross-over cable to connect them, thus alleviating any outside influences.
So basically, yes, a better add-on NIC tends to be better than cheapo on-board NIC, but only when coupled with other better add-on NIC's and good in-between equipment, for most "home" (re: not traffic smashed) LANs.
If you are simply having issues with a particular PC failing with backups across a network, I would try changing some settings on the NIC before swapping out hardware left and right. Assuming it's a typicval 10/100/1000 on-board NIC with full duplex, their are a LOT of settings you can change to try and improve reliability. For example, turning off full duplex, turning off the TCP, etc offloading, flow control, jumbo frame size, and many other settings. You migh not get mazimum back and forth communication speed, but for most things (including LAN backup), you would never notice it anyway unless you are very familiar with NICs and have already "optimized" them previously..
