Interestingly, though swell and svelte are similar and have similar meanings, the words are unrelated:
swell - O.E. swellan "grow or make bigger" (past tense sweall, pp. swollen), from P.Gmc. *swelnanan, of unknown origin. The noun meaning "wealthy, elegant person" is first recorded 1786, the adj. meaning "fashionably dressed or equipped" is 1810, both from the notion of "puffed-up, pompous" behavior. The sense of "good, excellent" first occurs 1897.
from Online Etymology Dictionary
whereas svelte comes from:
svelte - c.1817, from Fr. svelte "slim, slender," from It. svelto "slim, slender," originally "pulled out, lengthened," from pp. of svellere "to pluck or root out," from V.L. *exvellere, from L. ex- "out" + vellere "to pluck, stretch."
So is swell instead related to swill and swallow?
swallow (v.) - "take in through the throat," O.E. swelgan (class III strong verb; past tense swealg, pp. swolgen), from P.Gmc. *swelkh-/*swelg-. Sense of "consume, destroy" is attested from early 14c. Cognate with O.N. svelgr "whirlpool," lit. "devourer, swallower." Meaning "to accept without question" is from 1591.
swill - O.E. swilian, swillan "to wash, gargle," from P.Gmc. *sweljanan. Meaning "drink greedily" is from c.1530; the noun sense of "liquid kitchen refuse fed to pigs" is 1553, from the verb.
American Heritage says that swill and swallow come from PIE swel-, "to eat or drink." It is also interesting that the Proto-Germanic versions of swell and swill differ by only one letter. Swell seems related to swel- in the sense that something that eats a lot would get bigger. Online Etymology lists P.Gmc. *swelnanan (the precursor to swell) as of unknown origin but it seems that the root of the first syllable at least should not be in doubt.
