Yes - the beer will continue to carbonate after the first week. It usually takes 3 weeks to reach a good carbonation level (with the appropriate amount of priming sugar).
What temp did you store the bottles at for carbonation? - it will move along faster at warmer temps.
Did you chill the beer before you opened it?
I'm not sure how much 1/3 of a cup of sugar weighs, but for a 1.5 gallon batch, you only need 1.5 - 2 oz to get a lot of carbonation (3 - 3.5 volumes). If you're over 2 oz, you might be inline for some bombs (though I don't really know how many volumes of CO2 you need to have to reach explosion)
See The Beer Recipator - Carbonation