There are many bands who failed to make it in America. Some of them though I believe really should, when others it's better than they remained unknown, and even better they remained unknown globally.
Well, to be more to the point, most of those bands were known during their time, but yet pretty unknown to younger generations.
Those bands I'll mention had several Top 20 Singles or/and Albums in many countries, yet they never made it in Top 20 in American Charts.
(In alphabetical order)
1) Beatmasters (Neither any Albums nor any Singles, made it in American charts)
Although Techno Music was never popular in US (except several teritorries, like Michigan), many Techno bands have made it in US Dance charts. Beatmasters didn't, although they had a bunch of Top 20 hits during late 80s-early 90s in various countries. They did many remixes for many artists as well, for example Depeche Mode's Behind The Wheel.
2) The Boomtown Rats (Highest Charting Positions in America: Album: The Fine Art Of Surfacing #103 (1979), Single: I Dont Like Mondays #73 (1979)
A Punk/New Wave band mostly famous in UK and Ireland, and in Canada and Australia by some degree. Yet they were never popular in America as a band, although their lead singer (Bob Geldof) is globally known as one of the two organizers of Live Aid .
3) The Christians (Highest Charting Positions in America: (Single: When The Fingers Point #29 (US Dance) (1987), No Albums made it in American charts)
Another one of the late 80s British Pop/Soul bands, but yet a good one. Although they had many hits in Europe, I believe they "had it" to make it in America as well, and I believe lead singer's voice (Garry Christian) sounds pretty Americanized. Since George Michael did it, why not them as well? Guess lack of promotion.
4) Cock Robin (Highest Charting Positions in America: (Single: When Your Heart Is Weak #28 (US Mainstream Rock) (1985), No Albums made it in American charts.)
Ironically, Cock Robin were Americans themselves. It's not uncommon for a non-American band to not make it in America, yet it's pretty uncommon for an American band to lack success in America, yet be successful in Europe. Cock Robin scored several Top 20 hits in many European countries (except UK, where they had only 1 hit there as well, particularly The Promise You Made), but they had only one minor hit in America. The one I mention above, which made it in #35 in Hot 100 charts as well.
5) Fischer-Z (Neither any Albums nor any Singles, made it in American charts.)
Although they weren't so much popular in their homeland UK, they were pretty popular in mainland Europe during early 80s, in countries like Germany, Netherlands or Portugal. Their sound was pretty similar to the sound of The Police, although they were way less successful than the latter. A very underrated band.
6) Secret Service (Neither any Albums nor any Singles, made it in American charts.)
A Swedish Pop/New Wave band, with a few Top 10 hits in Mainland Europe, during late 70s and early 80s. Well it's not a surprise since many bands from Mainland Europe, fail to chart even in UK. It would be even harder for them to make it in America. Yet those guys I believe they deserved more chances unlike the highly overrated Abba for example.
7) The Stranglers (Highest Charting Positions in America: Album: Dreamtime #172 (1982), No Singles made it in American charts.)
Perhaps, the most notorious example here. Pretty successful in UK and all around Europe, and definitely one of the most influential bands ever, but still unknown in America, despite all their efforts to make their sound more Americanized since Aural Sculpture. Anyway, all their albums with their first Lead Singer (Hugh Cornwell) are definitely masterpieces. A great diversity of styles.
8. Pulp (Highest Charting Positions in America: Album: This Is Hardcore #114 (1998) , No Singles made it in American charts.)
Although Britpop genre was a really lousy genre in my opinion, Pulp were better than others. They took thigns to another level instead singing guitar-based cheesy songs. They started also way before than most Britpop bands of that era, releasing 2 albums (yet without any success) in the 80s. Unfortunately they didn't make it in America, unlike overrated Britpop bands Blur and Oasis.
9) Ultravox (Highest Charting Positions in America: Album: Quartet #61 (1982), Single: Reap The Wild Wind #71 (1982)
Ultravox were very similar with another Synth-pop band from that era, particularly Wang Chung. Wang Chung went to be pretty successful in America, with 3 Top 10 hits (they lacked success in their homeland though), but Ultravox hardly made it in American Top 100. And by the way, Ultravox's lead singer (Midge Ure) was the second organizer of Live Aid concert, I mentioned in #2.
10) Visage (Highest Charting Positions in America: Album: Visage #178 (1980), No Singles made it in American charts.)
It's quite surprising to see, that Visage weren't even an one-hit wonder in America. Their song Fade To Grey was extremely popular all across Europe, making it into the Top 10 in many countries. Apart from this hit they had some other Top 20 hits in UK, but in other countries they're mostly known for their song Fade To Grey.