For a fee... you can subscribe to Maponics' Neighborhood dataset
While Maponics provides mostly GIS data, (eg. allowing one to pinpoint on a map the boundaries of neighborhoods and such), the simple neighborhood list is also available, I think.
Another commercial offering is Urban Mapping's
In you target particular cities/counties, there are plenty of free resources to be found, oft' in the .gov / .us sites, for specific cities and counties. Unfortunately aside from the difficulty of locating such resources (there doesn't seem to exist any practical directory for such local gov-managed databases), there is no standard as to the format in which the data is stored or the specific semantics of the data collected. Luckily, ZIP-code is rather unanbiguous, and he neighborhood concept relatively general (even though the neighborhoods themselves can be quite dynamic, with bot the introduction of new neighborhood names, and some minor shifting of boundaries).
The overall complexity of the task of compiling such databases, the long half-life of the data, and the potentially lucrative uses of such data, seem to explain why it is hard to find non-commercial sources.