Yes, but it takes more than removing most of the identifying information.

First, the precise browser version and OS can probably always be identified by checking for supported features, bugs etc. even if the extreme measure would be taken to remove the user agent string.

Add the screen resolution, IP, timing and request patterns (+) and we are all screwed.

(+) e.g. rule out users that are using other sites at the same time. Note that it would be possible to determine if a page is in the currently focused and visible browser tab and forward that information to the tracker.