The most accurate mechanical watches are called “chronometers”, and for this characteristic they obtain a certification from the COSC, an independent Swiss institute in charge of checking the quality of watches movements.
For the COSC, a mechanical watch is accurate if, at the end of the month, has a difference of 2-3 minutes ahead or back compared to the exact hour (for more details about tolerance threshold see the COSC web site).
So, do we have to renounce to have an accurate mechanical watch? Swiss institute say yes… but we must not give up! A good watchmaker can improve a watch precision regulating the balance wheel, the part which determines the rhythm of the mechanism. The possibility for the watchmaker to operate depends on the quality of the watch (on the technical characteristics of the balance wheel, for example), but a fairly good result, even not excellent, can be achieved with a simple reasonable-cost operation.
In case of greater differences of time indication, the delay (or, more rarely, the advance) of the watch may be the signal of a mechanical problem. In this case, it is necessary act as soon as possible in order to avoid neglecting a mechanical problem which can damage, more or less seriously, the movement.
Of course, this is applied just to mechanical watches. Quartz watches are well known to be extremely precise due to the very high-frequency oscillation of quartz, and delays or advances are not allowed at all… otherwise, that means the battery of your watch needs to be replaced or, in the worst case, the mechanism needs to be checked.