I've noticed a slight yet consistent increase in skill check while using a toolbox. Therefore, I'm assuming the answer is yes, even though none of the toolbox descriptions mention this at all.
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They do.
If I remember right, they quadruple the skill check odds, from 8% to 40%.
It's only 40 %? Feels higher.
40% every 1 second. So over the course of using the entire toolbox you're very likely to get more skill checks overall.