Is there a connection between すみません (the word used for an apology) and the negative form of すむ (to live (in a place)), or are they unrelated? Does it perhaps mean something like "I don't belong here" or "This is not my place"?
Hmmm Where are you guys getting your information from? Yuki seems to think it isn't related... Please when you are saying such definate answer the obviously conflict with a teacher then PLEASE QUOTE YOUR SOURCES since Yuki is very careful about her answers.
OK, I did my research... SUMIMASEN does come from the verb 済む （SUMU = to end) as in しごとが すみました。（SHIGOTO GA SUMIMASHITA = My work ended.) This SUMU is different from 住む (SUMU = to live). SUMIMASEN is a polite negative form of SUMU and means something like "does not end", but that does not make any sense. So SUMIMASEN is actually from the concept of KOTO GA MONDAI NAKU SUMU (things get done without any problem). The negative version of that is something like "I did something that didn't get done without any problem". Therefore SUMIMASEN... Very few Japanese people know the origin of SUMIMASEN. They don't even use KANJI for the original 済む and write 済みません that much anymore. SUMIMASEN is written in HIRAGANA 99% of the time. Another possible origin for SUMIMASEN is from 澄む (SUMU = to clear). That theory is from the concept, IIWAKE SHITEMO KOKORO GA SUMAMAI (= My mind does not become clear even if I make an excuse...)
Don't think too hard to understand the origin. SUMIMASEN, to me and a lot of Japanese people, is just a word. I appreciate your comments, though. This gave me new knowledge on the Japanese language!
And this is why etymology is a dual-edged sword... On the one hand, learning how a word or phrase originated can help to understand what it means and why it means it. But as George likes to point out, sometimes it just confuses the issue when you're just trying to learn to use the word or phrase.