I think this is a won't fix. The properties of the mysqli or mysql_result object are invalid if the connection establishment or the connection is closed. In case of mysql_result is something wrong happens.
The extension needs different design, new mysqli(...) tries to connect, if something wrong happens it should throw an exception, it doesn't now. The design can't be changed that easily in a minor release.

The intent of the connect_errno method is to determine if an connection error occurred...one would think. The programmer is expecting the chance a connection error occurred. Don't error-out if he asks.

$host = 'bad_host';// FORCE AN ERROR -- debug
$mysqli = @new mysqli($host, NTSM_USER, PWD_NTSM, NTSM_INSTANCE);
if ($mysqli->connect_errno) {
  do something
}

One would expect the code above to not annoy the user with an echoed warning. I have to use @$mysqli->connect_errno, but I have no return from the method call. Instead I receive 'Couldn't fetch mysqli in', why is a fetch even attempted with this method? Look at the local object and see if it contains an error status and return it.

I have a shopping cart object stored serialized in a database session. This object has a mysqli object as a property, which is left open for reuse.
I was getting the "Couldn't fetch" error every time a new Ajax call tried to access the mysqli object (with OOP) (unless this was a brand new session and the mysqli object had only just been created).
I think the problem is that the database connection is lost between Ajax calls. I have solved it for now by adding a destructor to the cart object, which does mysqli->close() (if the object exists - my cart only instantiates it if needed, not in the cart constructor). 
I haven't checked, but it might also be possible to solve the problem with __sleep. Or you could just be more ruthless and ensure that you never ever leave the mysqli object open.