As I understand it the 250,000,000 number refers to total streams of all tracks on the album[1], it does not refer to plays of the
entire album, so the article is wrong. The album is 20 tracks which means we're looking at 12.5 million streams per track. There were (and still are) many offers for a free Tidal trial, which were shared all over the internet at the time the album became available, so I do think it's possible that Tidal could have had those numbers. Paying subscribers? No, but users? You could sign up to Tidal for free, and Kanye albums are very highly anticipated... Is it reasonable to expect ~3 million people to each listen to an album ~4 times through on average? I certainly listened to it
at least a dozen times in the first 2 weeks and know many friends who did the same.
For comparison Taylor Swift's album Reputation did ~500,000,000 streams in the first few weeks after it launched, granted that was available on all services and not just Tidal: http://www.bbc.co.uk/news/entertainment-arts-42564917
I'm a little suspicious of the numbers, given Taylor Swift has a wider appeal than Kanye and her album was on all services but you could sign up for Tidal for free and the article is clearly wrong when it says "a claim that would have meant every subscriber played the album an average of eight times per day" so I'm inclined to say Tidal are telling the truth because if they did inflate the numbers then they would not have needed to inflate them from, say, 25 million to 250 million which is what the report seems to suggest.
[1] See the BBC Taylor Swift article for an example of how the figures are calculated