A four inch router bit? Most router bits don't go that deep. And if one did, it would scare the crap out of me. Safest to drill it out and use a chisel to clean up.
However, you do realize that this is considerably more expensive, possibly more dangerous, more time consuming, and will result in more structural weakness than traditional joinery?
There's a reason most woodworkers don't build stuff this way, and it's not because of the glue. Wood is not like a piece of aluminum or plastic. You have to consider the grain and wood's weaknesses. Wood is not strong in every dimension like metal is. Wood changes size and mass
daily because of humidity and temperature. It is still a living thing and you have to work with that.
I'd recommend reading a book on building boxes, like this one:
Amazon.com: Taunton's Complete Illustrated Guide to Box Making (Complete Illustrated Guide): Doug Stowe: Books
Anything Taunton publishes is pretty good. You can also pick up a copy of
Fine Woodworking at most news stands - it is an excellent magazine. Another excellent source is:
http://www.amazon.com/Tage-Teaches-W...2652901&sr=1-2
Not directly on point for box making, but it is an authoritative guide to joinery and excellent overall.
If you might want to commission some boxes, check your local woodworking store. There should be one in the yellow pages selling furniture grade lumber and tools. A lot of them have shops and you could have someone make the cuts for you on pro gear. It's surprisingly cheap for small projects. They'll charge around $2-$4 per cut, or prorate work by the hour, probably $20-$40. But if you make, say, 12 cuts in 30 minutes, you can have it professionally done for $20 or less. Also, those places are a magnet for hobbyists. There should be a bulletin board up with people looking for work. You can find someone to build there, or put up a card asking for help. Someone will.