The spatial resolution of drift-tube counters of the VENUS muon detector in the test arrangement for cosmic muons was shown to have two distinct components. The first component, which corresponds to about 96% of the detection efficiency, has a normal distribution with an rms spatial sesolution of about 1 mm. It is evident that the second component with about 3.6% of the total incident muons corresponds to events for which delta-rays generated inside the chamber wall are scattered closer to the sense wire or between the sense wire and the true tracks. Such confused track events are unavoidable in drift-tube counters with a single TDC scaler per cell. The general characteristics of phenomena caused by delta rays in drift-tube counters are studied by Monte Carlo simulations for various detector arrangements. The upper limits of the knockon electron yield and confused track rate are found to be respectively 14% and 6% of the total incident muons with medium energies of the order of 10 GeV or less in typical muon detector systems consisting of aluminum drift-tube counters and iron absorbers. For drift-tube counters with very thin aluminum walls these limits are reduced to about 11% and 5%, respectively.