The band was formed in New Jersey in the early 1980's, and named after a Yosemite Sam expression ("Varmint, I'm a-gonna blow you to smithereens!") Amazingly, the original lineup of Pat DiNizio (songwriter, guitars and vocals), Jim Babjak (guitar), Dennis Diken (drums) and Mike Mesaros (bass) stayed together for 25 years, until Mesaros quit the industry to raise his kids.
"A Girl Like You" was the band's biggest hit, from 1989's 11 (named after the famous Spinal Tap amplifier). The song peaked at #38, while the LP hit #41. The Smithereens had a second top 40 hit, as well -- but it probably isn't the one that you are thinking of. "Too Much Passion," from the Blow Up CD hit #37 in 1992.
While the band's period of major success was relatively brief they have remained a working band, and continue to perform and record to this day.
Cool trivia fact: "A Girl Like You" was originally written for the Cameron Crowe movie Say Anything... Apparently, Crowe felt that the song gave away too much of the plot, so he did not include it, but he remained on good terms with Pat DiNizio (who did a cameo in the Crowe film Singles in 1992).