Not always. In my experience, it is more common for either
the primary to open, OR to develop a partial or full short.
And plate secondaries can go either way.

I just recently saw 2 quite different plate transformers
from different tube amps, develop dead shorts across their
high-voltage secondaries. (Both amps were Chinese made,
sorry to say...)

One way to help minimize this is simple: put a 2 ohm to 5 ohm
10W resistor in series with the primary. It limits the
huge inrush current at turn-on, helping prevent internal arcing.

Also a 100 ohm resistor in the plate winding, BEFORE THE RECTIFIER,
can help protect the secondary. Two 50 ohm, one on either side, if
it is a center-tapped winding with the CT grounded. Yes, even if
you have tube rectifiers.

These are simple and cheap things you can do, which mfrs.
usually don't--to save a few dollars.