The visual system can encode many stimuli across the visual field simultaneously, but the number of objects that can be fully identified in parallel is limited. At the extreme, some objects might have to be identified serially. One useful tool for distinguishing parallel from serial processing is the redundant target paradigm, which compares responses to displays containing one target to displays containing two targets. Many parallel models predict a positive redundant target effect: faster correct responses to two targets. Here we revisit the redundant target paradigm by developing and testing predictions for a standard self-terminating serial model that accounts for the accuracy as well as the speed of each response. Surprisingly, it predicts slower responses to two-target displays than one-target displays. To test that prediction, we conducted three experiments that each measured performance for three different judgments of written words: color detection (detecting colored letters), lexical decision (detecting real words among pseudowords), and semantic categorization (detecting nouns that refer to living things). In all the experiments, only the color detection task yielded a positive redundant target effect, while the lexical and semantic tasks yielded zero or negative effects. These results are consistent with low-level features (color) for two stimuli being processed in parallel, while the meanings of two words are processed serially. Altogether, this study informs models of reading and furthers the development of general theories of response time that include errors.