The simple answer is that any scoring, nick or other damage to the wire or insulation is unacceptable in that it creates, or increases the probability of, a potential failure. The two most common failures to worry about are localized heating and/or complete break of the wire. Increased heating might eventually cause a fire. Thermal expansion and contraction and/or vibration leading to a completely broken wire could, in theory could cause a short with more heating. For any wiring that is installed under the National Electrical Code, or many other code(s) that potential failure point will cause the wiring to not meet the code. An inspector that sees the initial damage will probably require that the wire be removed and replaced or repaired in a code acceptable manner. For the NEC that would require splicing the wire such as by use of an approved connector and possibly adding a new junction box, or complete replacement.
But that really doesn't answer your question, particularly because the answer depends on the standard(s), if any, the wire type and extent of the damage. There are non-code wiring situations where there is a low probability of a failure during the life of the device the wire is installed in, and if it does fail it might be acceptable. Just one of many examples is low voltage, limited current wiring inside some electrical device, for example a metal flashlight. If the wire is stranded and only a few strands are nicked, in theory it could eventually fail but if the current supplied by the power source is limited so even if the broken wire causes arcing or a short, no dangerous heating will occur. Of course the device wouldn't work and presumably it would be noticed and the problem repaired or the device thrown away.
The bottom line is that without knowing the details of the wire and the application it is hard to say what is acceptable.
