Informally speaking, presuppositions are meaning components which are part of the common ground for speakers in a conversation, that is, background information which is taken for granted by interlocutors. The current literature suggests an immediate processing of presuppositions, starting directly on the word triggering the presupposition. In the present paper, we focused on two presupposition triggers in German, the definite determiner "the" (German "der") and the iterative particle "again" (German "wieder"). Experiment 1 replicates the immediate effects which were previously observed in a self-paced reading study. Experiment 2 then investigates whether this immediate processing of presuppositions is automatic or capacity-limited by employing the psychological refractory period approach and the locus of slack-logic, which have been successfully employed for this reason in various fields of cognitive psychology. The results argue against automatic processing, but rather suggest that the immediate processing of presuppositions is capacity-limited. This potentially helps specifying the nature of the involved processes; for example, a memory search for a potential referent.