Note I used Gmail-app, Outlook-app and K-9. I am sorry, don't want to offend the developer at all...
Out of the above 3, Aqua Mail is by far the best in my opinion.
But that said, I never had these problems before, only with Aqua Mail it pops up, once in a while.
Then this go fine for a while, but like today, it pops up again and it looks quite stubborn this time...
I am not the developer, but just an Aquamail user and I consider myself being rather proficient in e-mail and networking (having had long experience with maintaining internet servers).
I have some thoughts about what is likely happening, based on what @federacer has written as the response from Alice support:
Finally I get a reply from Alice/TIM ... If you have TIM as your mobile operator, the mail works with apps like Aquamail or others ! If you have another mobile operator (I have Vodafone) the mail works only from browser (webmail) !
I suspect the reason for what you are observing is as follows:
Alice closed access to their e-mail servers from any mobile networks except TIM's.
So, if you are using any other operator, you would not be able to connect to the e-mail server from Aquamail. I suspect you can connect from the network that is not identified by Alice as "mobile" (i.e. when you are connected via Wi-Fi).
The reason why other e-mail apps are still working on those networks is probably because they are relying on the cloud services. Gmail and Outlook apps use cloud-based services where they pull and store the messages. Since those cloud servers are not on a mobile network, Alice is allowing them to connect to their servers. What I am still puzzled by is how K-9 can connect.
Sorry, no offense intended, but I suspecting that when you were testing this in K-9 you were not on non-TIM mobile network. If you still have K-9 mail app set up for your Alice account, you can test this by trying to check your mail from a non-TIM mobile network.
I hope this helps.
Kostya, the sole developer of Aquamail might have better insights in this when he gets to answer this.