You could say that the Indian States which ran public post offices were in competition with the Imperial, and later independent Indian, post offices.
Examples of usages of British Indian stamps purely within States which were running their own post offices aren't very common. This is because the State post offices were usually cheaper and/or more efficient than the Imperial post office, but here is one example, used entirely within Barwani State


This was a registered letter. While I don't have an example of a registered letter covering the same two towns at the same time from Barwani, this rather battered registered cover


shows that registered mail in Barwani using the State Post Office, two years later in 1926, still only cost 3˝ Annas - or ˝ anna less than the Imperial mail for the same service.
So the two post offices were in competition with each other - and choosing the more expensive Imperial PO was a distinctly eccentric choice!