Independent pieces of corroborating evidence should provide stronger support to a hypothesis than dependent pieces of evidence. Overlooking the inferiority of dependent relative to independent items of evidence can lead to a chain reaction of double-counting evidence, over-estimating the probability that the fact under consideration is true, and making wrongful decisions. Within fictitious scenarios, we investigate people's sensitivity to the independency advantage. We assess their ability to integrate multiple items of evidence that come from (in)dependent sources who differ in reliability. We find that participants properly perceive dependencies when explicitly asked but fail to distinguish the probative value of dependent versus independent evidence in their belief updating. Still, individuals who perceive a strong dependence between sources treat the evidence as being more redundant. We find no dependency-related effects on participants' individual Bayesian network model predictions. Potential reasons why participants perceive (in)dependencies and yet (mostly) fail to discount for them are discussed.