We analyze periods with elevated >40 MeV proton intensities observed near Earth over a time span of 43 yr (1973–2016) that coincide with the passage of interplanetary (IP) shocks. Typically, elevated proton intensities result from large solar energetic particle (SEP) events. The IP shocks observed during these elevated-intensity periods may or may not be related to the origin of the SEP events. By choosing those cases when the shocks can be confidently associated with the solar eruption that generated the SEP event, we analyze the components of these SEP events that are localized in the vicinity of the shock (so-called “energetic storm particles”, ESPs), focusing on those events where the ESP component exceeds 40 MeV. We examine the interdependence of these high-energy ESPs with (i) the properties of the solar eruptions that generated the shocks and the SEP events, and (ii) the parameters of the shocks at their arrival at 1 au. The solar eruptions at the origin of the shocks producing >40 MeV proton ESP intensity enhancements are within ±50° longitude of central meridian and are associated with fast coronal mass ejections (plane-of-sky speeds ≳1000 km s−1). The ESP events with the largest >40 MeV proton intensity increases tend to occur when there are structures such as intervening IP coronal mass ejections and other unrelated shocks present in the solar wind through which the shock is propagating. Among the various local shock parameters considered, only the shock speed shows a certain degree of correlation with the observed ESP intensity increase.