If you connect other mail accounts to Gmail/Inbox, Inbox will NOT warn you if it looses connection. When this happens in Gmail, you will be greeted with a big yellow warning.
My private mail server were down a day and Inbox apparantly never reconnected to it. Fast forward 10 days and I missed an important mail until it was too late, because I never noticed Inbox were disconnected.
I love Inbox and how easy it is to keep my mailbox clean, but due to this and also because I can't customize signatures to my different mail accounts, I went back to Gmail for the time being. The fact that it hides the signature also annoys me because I always HAVE to make sure it's there by expanding the [+] sign... I simply don't trust it :)
Don't use gmail....
Run your own mail-server, then your limits on email will be how much disk storage space you wish to purchase.
Email as designed is peer-to-peer technology. In fact, it was the first peer-to-peer system for the internet. Return to that world and all your troubles with the "central authority" (google) will disappear.
This is like telling someone who got poisoned by E. coli-contaminated spinach that the solution is trivial, just grow your own spinach, farming was designed as a means of small-group subsistence, return to that world and all of your problems with agribusiness will be over....
I love being a sysadmin and learning how everything fits together, and even I find it at times exhausting keeping up with everything related to email best practices.
For someone without any interest in such activities, it would probably be a royal pain in the butt.
I'd rather spend time with my wife then hacking on a mail server, but I still want to support open interoperability.
I ran my own email server for probably 20 years, starting with UUCP over USR Couriers. It got more and more difficult to keep things running, and I had better things to do with my time.
Every year the process gets more and more involved. Usually it's a new 'trick' or nuance that is needed to have Google accept my mails without marking them as spam.
It's not like the solutions are necessarily particularly hard (or bad, for example they made mandatory things like SPL when when IPv6 is used). But you end up with zero information to debug on. If you're lucky the message will bounce, but worse it just festers in your recipient's spam folder.
That said, I end up with something better -- SpamAssassin works great, with far greater success than Google makes of my employer's inbox (probably it has more cpu cycles per-message.) dovecot's IMAP implementation is far superior for searching and performance. Google's filtering is so bad it's of very limited use, procmail is much better. And you get to avoid Google's tiresome de-duplication of messages (a great help for mailing lists)
For about 15 years we administered a personal email server as a small co-operative of 3-4 people. This is the way to go IMO.
This isn't specific to Inbox; we have limits in place for all accounts (documented here: https://support.google.com/a/answer/1366776).
You can probably understand why this is necessary to keep our systems healthy--while the vast majority of users never encounter the limits, if they didn't exist, there's a risk that abnormal usage could limit our availability for every other user.
That said, I think the request to make this more obvious to the user is a really good point and something I'll look into.
A few questions: Is Inbox actually less secure than the GMail app? Is there a reason why using Inbox (an official Google app) doesn't prevent these messages from being sent? And will these e-mails ever stop?
I don't recognize that exact notification text, but I suspect you're seeing the notification mentioned here (https://support.google.com/a/answer/6260879). As noted on that page, this is triggered by use of password-based auth (rather than OAuth). Is it possible you're signed into your account from Apple Mail using just a password?
I found this out by breaking a recurring task on a site of mine, which had error logging set to email me. But the task was spinning through a a list of things to process, and throwing/emailing for each item (to a gmail label I only check occasionally). Then trying the whole thing again on the same (now somewhat bigger) list a minute later. It got up to a few thousand mails per minute before Google said enough.
I happened to notice it right away and put the fix in place, but it still took the better part of a day for things to percolate through and go back to normal. I don't think I missed any real mails from real people. They just arrived a day late.