Unfortunately it is not only the NMR tube but also some components in the probe that contribute to the background signal. There are several ways to reduce or eliminate the background signal.
1. There are pulse sequences that will discriminate between broad and sharp signals, mostly based on T2 relaxation. This requires lines of your analyte that are narrower than the background.
2. Take a spectrum of a NMR tube with the same solvent but without your analyte and subtract it from the spectrum with your compound.
To complement Clemens answer, I would like to mention backward linear prediction, which is a standard procedure in almost all NMR processing software packages. It helped me a lot for measuring 11B NMR spectra using standard (non-quartz) NMR tubes.