"Can't Happen Here" came when the band was firmly seeking commercial success. The song was off the 1981 Difficult to Cure album, which received mixed reviews (to put it kindly). Ironically, this more mainstream sound did not generate the huge success that Blackmore had hoped for; the album peaked at #50, while the single did not chart on the main charts in the U.S. (It did hit #13 on the Mainstream Rock charts).
The video is a pretty standard performance piece with some nice additional footage thrown in. The video features Blackmore and new lead singer Joe Lynn Turner (who replaced the awesomely named Graham Bonnet during the recording sessions for the LP). Turner would remain with the band until their 1984 breakup.
Rainbow would continue to focus on commercial success in the early 1980's, and would even score their one top 40 single, "Stone Cold" (which reached #40) in 1982. Readers may also remember "Street of Dreams," which hit #60 on the charts the following year.
Blackmore remains with the reunified Deep Purple until 1994, at which point he re-formed Rainbow. This version of the band released one album in 1995, which did not chart. Since the late 1990's Blackmore has been performing as Blackmore's Night with his wife (Candice Night); they are a folk-rock act.
Cool trivia fact: "Can't Happen Here" was the 52nd video played on MTV.