Yes mice can easily climb out of a bucket if there is not something slimy on the sides of the bucket to prevent the mice from climbing up.
You should coat the sides on the inside of the bucket with something such as olive oil, cooking oil etc and add a bit of water to the bottom of the bucket.
That way when the mice get onto the bucket mousetrap they will fall in and eventually drown in the bucket.
The cooking oil will prevent the mice from getting a grip and climbing the insides of the bucket so they cannot get out.
Then they keep falling into the water after trying to climb out of it and eventually they give up and then die in the water.
When making the bucket mouse trap you should also add some cooking oil or any oil to the spinning part of the mousetrap to make it slick so that the mouse cannot hang on and will fall right into the bucket.
You can also use some regular motor oil or even used motor oil to coat the inside of the bucket which makes it slippery.