We have investigated the solar wind drivers of magnetic storms during the rising phase of solar cycle 23 from January 1996 to December 1999. We used observations of coronal mass ejections (CMEs) by the Large Angle and Spectrometric Coronagraph instrument on SOHO and in situ solar wind observations by Wind, IMP 8, and ACE spacecraft. The storms were determined from both the Dst and Kp indices, and the study was limited to storms with Dst ≤ -50 nT or Kp ≥ 5. We show examples of different behavior of Dst and Kp indices during magnetic storms caused by different types of solar wind drivers. Furthermore, we have investigated cross-correlation between peak Dst and Kp values of storms organized according to the associated solar wind driver. It makes a difference whether a sheath region or the following ejecta causes the storm. We found that almost all intense and stronger magnetic storms (Dst ≤ -100 nT, or Kp ≥ 7-) were associated with shocks and CMEs, but for moderate storms, driver statistics were different in different phases of the solar cycle. We found different behavior of the Kp and Dst indices during different types of solar wind drivers. Intense and short-time disturbances, like postshock streams and sheath regions, generated more Kp storms, and ejecta generated more Dst storms. Thus one should be careful when comparing studies based on any single activity index.