I investigated briefly, and as it turns out the answer appears to be: yes, but with only a distant connection to the branch of the family running the bank, now five generations removed.
The hint for how to track this down came from a mention in the book Reflections on the Work of C.A.R. Hoare (Springer, 2010) that C.A.R. Hoare was of a "somewhat upper-class" background, with a footnote pointing to thepeerage.com, a site primarily focusing on genealogy of the British upper classes.
From this, I was able to trace his family history back along the paternal line as follows:
At this point the genealogy seems to connect to someone who has additional biographical details in Wikipedia, assuming that the Charles James Hoare is the same one described in this article (besides the same name, the approximate timeframe and occupations match). Wikipedia describes Charles James Hoare as "third son of Henry ('Harry') Hoare, banker, of Fleet Street, London, a Partner in C. Hoare & Co", so there's the connection to the banking family, at a great-great-great grandfather (five generations back) distance, around the late 1700s or early 1800s.
The generations after that appear unrelated to banking though: clergyman, clergyman, military officer, colonial civil servant, and finally professor. (The colonial-civil-servant occupation explains the somewhat well known fact that C.A.R. Hoare was born in Sri Lanka.)
Not that this distant connection is necessarily very important biographical information, but there you have it, assuming I haven't made an error in reading the sources, and assuming that they also haven't made any errors.