Nope that is not true in the general sense, which can be demonstrated simply by running:

open FOO, ">x";
$foo = <>;
[download]

in one window and then in another:

[jonathan@orpheus test]$ /sbin/fuser -u x
x:                    5665(jonathan)
[jonathan@orpheus test]$ mv x x1
[jonathan@orpheus test]$ /sbin/fuser -u x1
x1:                   5665(jonathan)
[download]

However it is true for most Windows applications.

/J\