Generally, if you're given evidence for something, the evidence-giver is trying to convince you of that something. If you're given only weak evidence, that itself is evidence that there is no strong evidence (if there is strong evidence, why didn't they tell you that instead?), and so in some circumstances it could be rational to downgrade your probability estimate.
When I read the title I had expected that this was the point of the post. Perhaps because I've been intending to write a post to that effect for the last three years or so.
So I agree completely.
Article: Weak supporting evidence can undermine belief in an outcome
Paper: When good evidence goes bad: The weak evidence effect in judgment and decision-making
Abstract: