Picking marbles

You can look at your urns as an array of 4 bit integers:
$0001_b$
$0010_b$
$0011_b$
...
$1111_b$

On every step you can set one bit to $0$ on every integer for which it isn't already 0. There are 4 bits so you can do it in 4 steps. If we go back to decimal, you're removing 8, then 4, then 2, then 1.

In fact we can also prove that $n$ is the minimum number of steps for $n$-digit urns through a recursion on the number of digits.

It is possible in 4 days:

First day you reduce the number of balls by 8 in urns with at least 8 balls. So now each urn has at most 7 balls.

Second day you reduce the number of balls by 4 in urns with at least 4 balls. So now each urn has at most 3 balls.

Third day you reduce the number of balls by 2 in urns with at least 2 balls. So now each urn has at most 1 ball.

Last day you took balls from all the nonemty urns.