Hi, I am posting this for the benefit of all the people who receive the ineligible account error when trying to play a PS4 EA game (like Need for Speed in my case).
After three live chats and 4 hours of back and forth. they finally found the solution.
The problem was this: one time in the past I had download a demo of an EA game on my PS3. It prompted me to create an EA account to play it, which I did. The game did not catch my attention and I deleted it from my PS3. As I started to receive advertising from EA on my email, I requested to delete my account. After that deletion was confirmed, I deleted from my email inbox all traces of that conversation.
Fast forward some months, even a year, I had forgotten about all that. I bought a PS4 and bought an EA game in the Playstation Store. I am not a big gamer, so this was actually the second EA game I tried in my life. When trying to load the first screen, it throw me an error that I did not have an EA account or something. And the game prompted me to CREATE a new EA account. Which I did. Using the same email address I had used the previous time, which is the only email address I have ever had with PSN. Turns out that by doing that I had now two accounts with EA. One that was "deleted" but was linked to my PSN account and another that was active but could not be linked. It is the one I am writing from right now.
I was told that my PSN ID that I tried to link to my active EA account was already linked to another account that was probably deleted and that I could bring back to life if I made a "forgotten password" request. I tried with all my active email addresses. With two never received any reply from EA. With the active one, I did, changed my password, but that didn't resolve the situation. I was still unable to link my EA account to my PSN ID, even though both shared the same email address!
After four hours and I was just despairing, I happen to mention to the support guy that I had never used any other email address with my PSN. So he proposed the following: he would change my email address in my active account to another of my emails while we were doing the live chat, and I would try to recover the password of my original email address (the one I had logged in for the live chat). And amazingly, that worked. Turns out that the EA account ID or gamertag is unique, and it is linked forever with the PSN ID, but the email address field in their databases is not unique! So they can have two EA accounts with the same email address! So, success. This EA account I am writing is just for telling you not to despair. Except... woe to you if you have ever created an EA account and linked it to your PSN ID with an email address that you don't have access anymore. That will be, I guess, unsolvable.
Good luck!