With piston ring technology these days, rings should seat almost immediately. There are exceptions, such as chrome rings. I am assuming you are using factory rings? As long as you aren't breaking in an engine with synthetic oil, they should be good right off the bat. After a couple of runs, you can then switch to synthetic oil.
Next question(s): did you just re-ring the engine with no other mods? I am curious as to the shape of the cylinders. if they aren't true, then you might have a seating/sealing problem. If you bored out the cylinders that shouldn't be an issue. Did you measure the bores for taper and out-of-round? If you did, and they are straight then I would be looking at valve sealing. Do you have the ability to check cylinder leakage? That way you can test it without taking it all apart again.
Another simpler thought might be to verify your valve lash isn't too tight?