Comprehension deficits that typify adults with right brain damage (RBD) have been linked to considerations of processing capacity and processing demands, as well as to ineffective suppression of mental activation that is incompatible with a contextually intended interpretation. As a first step in investigating how processing resource factors and more specific difficulties like suppression deficits interact to yield characteristic RBD comprehension patterns, the current study was designed to assess whether suppression function consumes attention. A total of 28 RBD and 22 non-brain-damaged adults listened to sentence stimuli that biased the meaning of a sentence-final lexical ambiguity (e.g., “spade”). The suppression task involved speeded judgements of whether a subsequent spoken probe word fitted the overall sentence meaning. In experimental stimuli, the probe word (e.g., “cards”) was unrelated to the biased meaning of the ambiguity. Comparison stimuli ended in an unambiguous word (e.g., “shovel”) that was clearly unrelated to the spoken probe. Thus, slowness after an experimental sentence, relative to its comparison sentence, indicated that the contextually inappropriate meaning of the experimental ambiguity interfered with the probesentence relatedness judgement (i.e., had not been suppressed). In two dual-task conditions, participants allocated 20% or 50% of their “brain power” to a concurrent secondary task, reporting orally whether the probe word consisted of one or two syllables. For both groups, suppression of contextually unintended meanings of lexical ambiguities was more effective in a single-task condition than when attention was shared with a secondary task. The secondary syllable-counting task also suffered when allocated less attention. Effective suppression consumes finite processing capacity. As elaborated in the paper, several combinations of these variables could underlie relatively good and poor comprehension after RBD. Researchers and clinicians need to keep in mind such potential interactions of ineffective comprehension mechanisms, stimulus/task processing demands, and processing capacity.