Can tile be removed and placed again before it sets?
I see your question was posted two days ago, so without a bunch of discussion, I think the answer at this point is a simple "no", if the tile is still on the floor.

According to the instruction sheet for VersaBond tile mortar (a product typically used to adhere tile to floors), the tile is ready for light foot traffic after 24 hours. 

intructions here

In my own experience installing tile, removing a piece that has been set with mortar is the kind of thing that can be done if it's caught right away, and if it's only a piece here or there. Even if the decision to remove half a room's worth of floor tile was made immediately, say within 5 to 10 minutes of setting, doing so would take time, meanwhile the mortar would continue to harden on the the tile, including the backs of the removed tiles and the sub-floor where they were removed. Cleaning that mortar off the removed tiles and the sub-floor before it hardens to a point where it needs to be chiseled (effectively ruining the tiles, plus making a big chore out of preparing the floor again) is the main problem with removal.

It would be considered an emergency in the scope of the job. And if discovered mere hours after the fact, I would not consider it as an option but would discuss the issue in terms of "starting over" with new materials.

Good luck!