Examining personality traits as predictors of human behaviour is of high interest. There are several but inconclusive reported relationships of personality and the susceptibility to the "anchoring effect", a tendency to adjust estimates towards a given anchor. To provide an answer to variably reported links between personality traits and the anchoring effect, we collected data from 1000 participants in the lab and validated typical anchoring effects and representative personality scores of the sample. Using Bayesian statistical data analyses, we found evidence for the absence of a relationship between anchoring effects and personality scores. We, therefore, conclude that there are no specific personality traits that relate to a higher susceptibility to the anchoring effect. The lack of a relationship between personality and the susceptibility to the anchoring effect might be due to the specific anchoring design, be limited to specific cognitive domains, or the susceptibility to anchors might reflect no reliable individual cognitive phenomena. In the next step, studies should examine the reliability of anchoring effects on the individual level, and testing relationships of individual traits and anchoring effects for other types of anchors, anchoring designs, or cognitive domains.