During a full round action such as the Withdraw Action, can one move full speed, slow down to half speed to tumble (i.e.) and then pick up their speed again?
I had thought you break it down to the equivalent of two move actions, where you set your speed as either half speed, or full speed. One player insisted that as a base 30', he could go 10 squares, by simply speeding up again in the middle of a move.
I think you're thinking about it the wrong way. You don't decide whether or not to move at full or half speed for the entire action; Certain actions just require you to spend more movement.
Last week my wizard needed to tumble her way to safety to cast a spell. She was hasted, by the way, so her total speed was 60ft. She moved through three squares (15ft) threatened by enemies, which cost her 30ft of movement. The rest of her movement was unhindered, so she was free to move the other 30ft normally. As a result, she gracefully tumbled away and killed everyone with chain lightning. =)
If she had used the withdraw action (without haste), she could have still moved up to 60ft. The differences are that the first square would not have been threatened, but the other two would have been (costing only 25ft of movement instead of 30ft); and that once she got away she would not have been able to cast that extremely helpful spell (since withdrawing is a full-round action in most cases).
So, the short answer to your question is... Yes, you can. If just depends on your surroundings, and if it forces you to move at half speed.
Another easy example is if you had to withdraw and move through a grease spell. You can move up to twice your speed (60ft in most cases), but you would still have to count each square of grease that you moved through as 10ft instead of 5ft.