randomstring: You're correct in assuming that the value the tax assessor assigns to your property shouldn't generally be more than you're paying for it. Exceptions are things like sales between immediate family members, sheriff's sale and other types of distress sales. The FMV
of a property is supposed to be set by what a willing buyer would paying a willing seller, etc. etc. etc, as it is with the price of most things in an open market. In your case, if you don't fall under the distress sale exceptions, and assuming you didn't get financing through the seller (this could be argued to have a value), then you have good evidence that the market has spoken on the value of your property, i.e., that what you paid should be FMV.
It's generally recommended that before filing for an appeal, you set up a meeting with the tax assessor in order to try to resolve the issue less formally. Ask to see the property record card for your property. It lists the age of the house, # of baths, sq. footage, etc. Look at yours to determine if corrections need to be made (do they list you as having a finished bsmt. or A/C which you don't really have, do they claim you have 3 baths when you have 1.5, and so forth). If this is the case, the assessor can agree to make an adjustment w/o an appeal. All a matter of negotiation w/the assessor at this point.
If you can't resolve it that way, filing for an appeal will cost ~$100-125 IIRC. But please take note of what I'd call a pitfall re tax appeals in NJ. The law allows the assessor a 15% (!!!!!) margin of error. (As I'm sure you know by now, this can translate to a lot of $ in our state.) An error not exceeding that will automatically lose on appeal. On a brighter note, there is a singular exception to the 15% rule. In the year of a reval, it doesn't apply. Downside to the exception is that e.g., in my town it has been 18 yrs. since the last reval. Some towns do them more often, so check w/yours.