You may think many users automatically prohibit the sending of push notifications and that may be the case for those who feel push notifications are intrusive and annoying. That being said, research actually indicates that app retention and engagement are up In fact, 68% of users opt-in to push notifications according to Responsys' research (http://www.responsys.com/blogs/nsm/mobile-marketing/push-notifications-marketers-new-power-tool/ ). Furthermore, research shows that engagement is up because of push notifications (http://www.emarketer.com/Article.aspx?R=1010950). . That's why marketers are focusing in on push notification strategies according to Localytics research http://www.emarketer.com/Article/Proof-App-Push-Notifications-Actually-Matter/1011149.
So if you know of an app getting poor engagement over time, the question really should be "Why is our app not engaging and what would engage users?" Also, "Is it obvious what benefit someone gets by opting into the push notifications?"
Push notifications are a great way to engage app users, as long as it's not annoying. One example of this is wayfinding apps with beacon technology that detect when mobile app users are nearby and send push notifications to let them know something important. In this case, push notifications could alert a user of a special sale, which can be perceived as spammy, or that they have arrived at a certain desired waypoint (more useful, and less spammy), along with interesting information about that location. Think of how useful these are for tourist destinations, like museums, zoos, etc.
You've downloaded xyz zoo app, walk near the big cat area and get a push notice, "grrrrowl, don't miss our feeding time. In 30 minutes you can see us chowing down on fresh antelope quarters".
Because it is annoying. Many people have various apps installed. If they enabled push for every app that would basically mean their phone ringing / vibrating almost every hour which is really annoying. They only enable push notification from important apps that are crucial for them such as Whatsapp or Twitter.