There's no chance of that; they limited notifications to 20 a month or so back. And even though you might get an alert showing there's more than that - like I did the other day - you'll still be limited to 20.
That's notifications of new content from artists you follow. I first noticed it last month when I was away for three days and had 58 notifications showing on my return but was only able to access 20. @SpotifyCares confirmed that notifications of new content are now limited to your latest 20.
I still don't recieve the notifications as i used to.
For exemple: I recieved a " album is out now" notification on my phone, but nothing happens on my desktop... it doesn't show a single notification. I mean, if we have the option to "follow" an artist it would be good if that function acctually worked =/
I've noticed that I generally have to shut Spotify down and restart it for new notifications to show up in the PC app. I get them via email okay, but there's definitely a delay or other blockage occurring on the desktop.
On the positive side, the links from the emails launch the albums in the PC app just fine. However, I have occassionally had issues with new albums I've been notified about via email not being found when I try to play them.
A particuarly curious one was the new Boris album "Noise" which launched and played okay, but which also produced an "album not found" error when I tried to click on the album name link! I saved it to my playlist, but can't actually access the album page itself.