In what used to be a regular ritual, every 4 years the Indian contingent used to return with zero medals (as in 1984, 88 and 92) or with one medal (bronze in 1996 and 2000, silver in 2004). Things began looking up in 2008 (one gold, two bronze) and in 2012 (two silver, four bronze) but normal service was resumed in 2016 with one silver and one bronze.
Anyone who has looked into the history of India at the Olympics would know that the first formal Indian participation was at Antwerp in 1920. They didn’t win anything there, but the hockey team won gold at Amsterdam in 1928 and did not look back for a long time.
(Footnote: Though India was not an independent country then, it did participate separately from Britain in several sports (such as cricket from 1932). A newsreel of the 1936 Olympics shows Dhyan Chand and others marching behind what appears to be the Union Jack.)
But the IOC records show that Norman Pritchard, who won two silver medals at Paris in 1900, did indeed represent India (when he was selected in Britain but participated as an Indian). He was the only participant listed as an Indian. Not much was known about him until recently. However, this article should serve the purpose. It even covers his later life as a second-rung Hollywood actor:
https://en.wikipedia.org/wiki/Norman_Pritchard
One might argue as to whether he should be considered as an Indian sportsman. But the IOC records do say that, so we may consider that to be the last word on the matter.
Update: an article from 2018:https://www.bbc.com/news/world-asia-india-44722759
abn397 
This was also a question in one quiz I participated! And Dhyan Chand and his men sang Mere Bhainse ko danda kyo mara when they won the gold and did not have a national anthem for India!
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You might see the marchpast on Youtube -probably it appears in Leni Riefenstahl’s famous documentary.
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