The move appears to be the result of two Bollywood production companies attempting to stop pirated copies of their films being viewed online within the country.
A government agency emailed the BBC copies of the court orders involved.
They list 2,650 websites that a judge ordered internet providers to block. Most are file-sharing services, but the Internet Archive is also included.
The San Francisco-based non-profit is best known for its Wayback Machine - an online tool that allows the public to see old versions of websites. It contains more than 302 billion saved web pages.
'Disappointed and concerned'
The website bans were granted by the High Court of the Judicature at Madras on 2 August.
They had been sought by Prakash Jah Productions - maker of the comedy Lipstick Under My Burkha - and Red Chillies Entertainment - creator of romance-themed Jab Harry Met Sejal.
Affected users are now being shown a message saying that access has been restricted under the orders of the government's Department of Telecommunications.
The notice had not explained the cause, which had led to confusion.
"Courts and security agencies do block certain websites and the reasons are sometimes not disclosed," Shambhu Choudhary, the director of the government's Press Information Bureau told the BBC.
The Internet Archive had earlier told the Medianama news site that it was also at a loss to explain the situation.
"Obviously, we are disappointed and concerned by this situation and are very eager to understand why it's happening and see full access restored," said office manager Chris Butler.
Although the blockage is reportedly widespread, some locals have reported still being able to access the archive.
India had 462.1 million internet users out of a general population of 1.3 billion people in mid-2016, according to the Internet and Mobile Association of India.
That makes its online audience the second largest in the world after China.
In 2014, India ordered local internet service providers (ISPs) to block the Internet Archive, along with Vimeo, the Daily Motion and 29 other popular sites, over concerns they provided access to "Jihadi propaganda".