Perhaps you can be more specific about the problem.
1 Profiles for each paper and ink combination?
2 Same settings in the printer driver and imaging application as when the profile test print was made?
3 (assuming you are printing from Photoshop) What rendering intent are you using?
4 Have you recalibrated the printer since the profile was made?
5 Are you using a third-party RIP?
6 What did you use to calibrate the printer
7 (assuming you made the profiles yourself) What hardware and software package was used to produce the profiles?
If you have, correctly made, dedicated profiles for each of the paper and ink combinations then you should not see significant colour variations between papers, unless something has changed between the time you created the profiles and the prints. All the profiles for inkjets using a 918 patch chart that I have made have been pretty accurate. However, even with correctly made profiles for the papers, you may see compression in the blacks when you select certain rendering intents. The best performance for blacks (from Photoshop CS) should probably be “relative colorimetric” with “black point compensation” checkbox selected. Although this will not be perfect, the tonal variation should be smooth (assuming that your image data is smooth).
If you have selected “absolute colorimetric” as the rendering intent you may see what looks like yellowish colour-casts in the print, but this often happens because the profile is (correctly) attempting to compensate for a strong bluish cast due to the optical whiteners in many papers. Absolute colorimetric would also (perhaps) compress the blacks because the printer and profiling system is attempting to match EXACTLY the data from your image file - which the paper/ink may be physically incapable of doing. “Relative Colorimetric” PLUS “Balck point compensation” (or some people prefer “Perceptual”) should be visually better.