That is basically what happened with Hulu. The service is owned by the major content providers, such as Fox and CBS, and all of the televised content is almost completely exclusive on Hulu now, as opposed to being on Netflix. As more and more shows started to get pulled from Netflix, Netflix had to rely on original shows to make sure that they have a full catalog of streaming content. They paid over $6 billion dollars in creating new shows for 2017.
Unfortunately for music, the exclusivity has started to show up in some cases. Artists that own the Tidal service don't allow any of their music on competing services, and Apple has been signing contracts with musicians to have new content show up on their service several months in advance before it does on other services (like Jay Z and Kanye).
It's been kinda the reverse in video, if my recollection of the timelines is right. Netflix's big push into originals and dropping of licensed content started to happen 4+ years ago, well before Hulu seemed to have the budget to get big shows into their back catalog.
From Netflix subscriber numbers, it's also not clear that the lost content has dented their subscriber retention at all. The talk among my friends centers much more around new Netflix originals (or even Amazon/Hulu originals) than around back catalog stuff on Amazon or Hulu. I don't understand why -- I'm utterly uncompelled by 95%+ of what they're putting out, since I don't have the motivation to deeply investigate new Netflix shows/movies every week, and however they're advertising, it doesn't really reach me -- but it's what I see happening, much to my annoyance.
Unfortunately for music, the exclusivity has started to show up in some cases. Artists that own the Tidal service don't allow any of their music on competing services, and Apple has been signing contracts with musicians to have new content show up on their service several months in advance before it does on other services (like Jay Z and Kanye).