For example, if a Pick and Place machine accidentally drops a component, will it realize this and grab an identical replacement?
The only reason I can think of as to why it wouldn't do this is to save time.
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The machine will detect failure to pick a part up to the point where the vision system sees it, but not usually afterwards. The normal process is for PCBs to be visually inspected, either manually or automatically as a seperate process after placement, to detect misplaced or dropped parts. Most P&Ps will probably not know they have dropped a part after it has been imaged, although it is possible some might sense placement pressure and/or vacuum pressure to detect drops, however if set up properly, drops shouldn't really happen.
Visual inspection is getting ever harder with parts becoming smaller and smaller. Note that an 01005 resistor or capacitor is only 0.4mm long, and a camera can't always tell whether the part is there or whether it's looking at the bare soldering pads.
Also: visual inspection after each placement is expensive. A P&P machine may place 10 to 15 parts per second, and decreasing this speed will increase production cost.
So checking after placement of individual parts isn't always possible. Most of the time the completed board is inspected, sometimes visually, either manually or automated, like Mike said. This check may not be conclusive, however, as the problem with the smaller parts remains. The proof of the pudding is the eating, so the best way to test is to see if the board works. This is done on a bed-of-nails, where power and signals are supplied to the board by test pins on a test-jig, and signals are measured at critical points. Instead of a bed-of-nails flying probe may be used, but the principle is the same. Microcontrollers are often pre-programmed with a test program which aids in this board testing, resulting in a go/no-go verdict.