Yep they can, but they're significantly less likely to bid for a 3* or 4* player, whereas managers who are the same level as you will definitely want to bid.
So if you can save some of the good 5* players with long auction times to your favourites list, you might get lucky and have the server unable to display those players later in the day. If you can then identify this, you may be able to get the players cheaply (compared to equivalent 5* players who don't get bumped off the transfer list).