I think everyone posted, replied under this thread (and many other stackoverflow threads) experienced the same issue. The issue seemed resolved for us just yesterday for a day, but today, it started acting up again, had to switch up wifi to log into Instagram. We also added email login after this incident.
at the time of my last writing, i was sure as all accounts are working fine. But today, it started acting up again. Now I had to change up to different wifi to login with Instagram.
We started having same issue since last Friday (Dec. 15, 2016) after working fine for months. Same user can login via oauth on one wifi, but not on the other, or even on mobile cellular network. And even when it worked on that wifi at one point, it does not work at other times. We have not found a workaround to fix this, reported to Instagram 24 hours ago, waiting...I tried to login to my account on instagram.com first, then go back to our site, browser would log in automatically. So far, we don't have a solution yet.