How to force iMessage on iPhone to download messages stuck on Apple's servers?
I have no solid reference to back this up, but I have seen it empirically many times over many years &amp; devices.

Once a message has been received &amp; read by your account, no matter what device, other devices will no longer receive that message.  

There may be a timeout period - as you can clearly see when you have your phone &amp; Mac next to each other, receiving at the same time. The one that last 'spoke' will be the one on which an incoming message will actually 'ping' &amp; the other will wait about a minute before also pinging...  

...But if one is off [&amp; this is the bit about which I have no real information ... how long it needs to have been off &amp; remain off], &amp; the message is received &amp; read on the other, then the first will never receive the message at all.

The same can be seen if your phone is outside reception range &amp; a message is received on the Mac [at home]. You only discover the message when you get back home.

See related: How to tell Messages &quot;I&#39;m not at home&quot;?