The role of the adhesive thickness in the failure of bonded joints is a controversial issue. On the one hand, analytical and numerical models show that the bonding strength is improved as the adhesive thickness increases, on the other hand experimental observations lead to the opposite conclusion. Taking advantage of the thinness of the adhesive layer when compared to the overall dimensions of the structure, a matched asymptotic expansions procedure allows modeling the initiation of a debonding in a single or double lap joint assembly. If there is a perfect adhesion between the joint and the substrates, the conclusion is identical to the first mentioned. In case of an imperfect bonding, i.e. if there is a micro-crack close to the end of the bonding zone, the trend is reversed and the conclusion meets the experimental observations. Special cases of thick and thin interfaces are highlighted as well as the sensitivity to flaw size.