Weeeeell....there's sort of an exception: but it's not really an "exception", merely the way the law is written. IIRC, it was the '68GCA that required handguns to have serial numbers and thus plenty of pre-1968 firearms (handguns and long guns) don't have serial numbers. They are lawful to possess. I'm sure I'm not alone in saying I have several in my own collection.
Likewise, there's no requirement that any post-1968 Title I firearm have a serial number if it's a homemade one - i.e., one you made for yourself and made without the intent to sell. Case in point being the so-called 80% AR15 and 1911 receivers that folks finish at home. You can make them and possess them lawfully. In fact, you can even sell or trade them as long as it's not your intent to make them for sale/disposition. (If you make these with the intention of selling/trading/gifting/giving them away you need to have an 07 FFL (Manufacturer's FFL.))
So while it's lawful to possess certain firearms made without a serial number, once one has been applied/assigned, it's unlawful to alter or remove the serial number.
The caveats to all of this are that: A. this is per federal law - state laws may vary; and B. Title II/NFA stuff has to have a serial number; that's how it's registered and tracked in the NFRTR. (I just registered a 12" H&R shotgun that originally had no serial number so I had to create/assign one to it. I chose "7/11CC" - it's short for "7 Eleven Credit Card"
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