I just saw the phrase “wiit und breit” (far and wide; in a wide area around) in a Swiss German caption to a picture, and that got me thinking.
High German ‹ei› /ae/ can come from (at least) the Old High German diphthong ‹ei› (however that might have been pronounced then?) or the long monophthong ‹î›, but not all German dialects have merged them: so that, for example, ‹zwei› < ‹zwei [neuter]› “two” and ‹drei› < ‹drî [masculine]› “three” rhyme perfectly in standard High German but not, for example, in Bavarian.
So I looked up both ‹weit› and ‹breit› in Grimm’s dictionary of German, since it also gives etymologies, and found that indeed ‹weit› comes from an earlier form ‹wît› while ‹breit› used to be OHG ‹preit› > MHG ‹breit›. (And indeed they don’t have the same vowel in English, either: ‹wide›, ‹broad›.)
So I suppose Swiss German also doesn’t merge those two OHG phonemes, which is fair enough.
I still found it interesting that even though those words don’t rhyme there, they’re still used in this phrase—presumably under the influence of standard High German.
Edit: This answer on wer-weiss-was.de says that the use of that phrase in Swiss German (and Saxonian, and other dialects) is a loan from standard German, which would have been my guess, too. (And apparently, in Old High German, the idiom was “wîd un sîd”.)