Looks to me as if your 4 'frames' are pocket screwed together. Are they glued, as well? How are the corners attached? What is that, 1x6" pine?
IF the four frames are glued as well as screwed, _AND_ the four frames are securely attached at the corners (glued, preferably) so that you don't get any racking / twisting movement, then you might get away without the 2x4's. Most of the weight is straight compression load, but every time you (or a kid) leans on it, you get fairly high levels of racking stress.
If, on the other hand, the only way your frames are built is with the pocket screws, no glue, and the panels are not securely glued into a stable box structure... or if the corners are only attached to each other by screwing into the 2x4's, then no, I would say that it won't hold.
Amount of weight going on the stand makes a good deal of difference too... Figure 10 lbs per gallon, at least. If you're setting a 10" tall frag tank on there, it'd be a dramatically different load from a 30" tall display tank. Planning a canopy that sits on the tank? Add that weight too...
If it can't stand several people standing on it... dancing 'The Twist'... then it probably should be stronger. A living room full of broken glass, water, and sand is a real nightmare to come home to.
If it's just a bit high in one spot, I'd take a rasp to it, make it level, and leave the supports.