It's mostly an issue with semiconductors. Counterfeits range from remarking the manufacturer from some knock-off with questionable quality control to a more reputable manufacturer to remarking part numbers with something that's similar-ish but with wildly differing specs to just flat out lying about what the part is. There was a major issue a while back with counterfeit Atmel ATmega328 parts (commonly used on the Arduino line) that were actually some random prototype device from ON semiconductor. The only thing they had in common was the package style and number of leads. Suffice to say, they did not work in the intended application...
It can be an issue with passives too, though. Remember the whole electrolytic capacitor issue? That was caused by some industrial espionage gone wrong, but it was made far worse by the fact that the defective parts were often represented as though they were more reputable brands (Nichicon, Panasonic, Nippon Chemi-con, etc.). I've also seen wild things like standard film/disc caps being re-marked as X2/Y2 safety rated, which is a major fire and/or shock hazard!
Digi-Key, Mouser, etc. are actual manufacturer authorized reps and buy directly from the manufacturer, not a 3rd party clearinghouse in China (or elsewhere; China is far from the only culprit, here). This tends to minimize the changes of getting counterfeits, and they'll make things right VERY quickly in the event you do. There's a reason mostly sourcing people grumble when they have to resort to buying from 3rd party clearinghouses overseas, and there's also a reason why design engineers grumble when CMs in China "tweak" the bill of materials to "use locally available parts".
Regarding buying parts made in the USA, I'm not even sure you can buy resistors made in the USA at this point. Probably some niche application ones, I guess. What matters is the control the distributor exerts over their sourcing chain. The part may be made overseas, but you have high confidence that the part is exactly what it's being represented to be. This is rarely the case with buying off e.g. eBay (then again, they rarely even bother to tell you who the manufacturer is in the first place).