Please explain a real world case on how to get both those messages at the same time

Can't replicate this at all.

@brianteeman - Go to article manager, select some articles and unpublish them.

Or you can inject the following HTML via your inspector, directly above the main <form>

<div id="system-message-container" class="j-toggle-main span10">
	<div class="alert alert-success">
		<button type="button" class="close" data-dismiss="alert">×</button>
		<h4 class="alert-heading">Message</h4>
		<div class="alert-message">2 articles published.</div>
	</div>

	<div class="alert alert-warning">
		<button type="button" class="close" data-dismiss="alert">×</button>
		<h4 class="alert-heading">Warming</h4>
		<div class="alert-message">oops, something went wrong</div>
	</div>
</div>

@C-Lodder I think this line might be a problem for some browsers
https://github.com/joomla/joomla-cms/blob/staging/media/system/js/core-uncompressed.js#L344
I propose to change it to:
messageContainer.offsetHeight = 0;

Only happens in Chrome.
Create an admin menu and one buttom,
The module for this menu, In the options: Check Menu in YES.
Now the admin show a message:
"Warning
The administrator menu Admin does not contain - Menu Manager, Module Manager, Components Container. Click to turn on the menu recovery mode."

Closed this message and the zone is fixed.
Ok for Explorer And Firefox

Not sure why considering the BS2 docs work perfectly fine.

@C-Lodder I don't have access to a 3.x install right now, but the problem is in Joomla.removeMessages.
Maybe messageContainer.offsetHeight = ''; or resetting the margin(s) ?

I'll take a look when I'm home. Have a hunch is something to do with the rubbish in template.js

check also the initial commit: #5257

My hunch was correct. template.js

It's all to do with that crappy sidebar that can slide in/out, which then changes some classes to manipulate the width. Enjoy fixing.

WON'T.
Let's introduce custom elements, alerts seems an easy starting point

Custom elements in 3.x?

Oops, thought it was for 4...

J4 we have fixed alerts ;)

In J4 we need to go away from bootscrap... ?

Closed as a duplicate of #12155