While scales measuring subjective constructs historically rely on agree-disagree (AD) questions, recent research demonstrates that construct-specific (CS) questions clarify underlying response dimensions that AD questions leave implicit and CS questions often yield higher measures of data quality. Given acknowledged issues with AD questions and certain established advantages of CS items, the evidence for the superiority of CS questions is more mixed than one might expect. We build on previous investigations by using cognitive interviewing to deepen understanding of AD and CS response processing and potential sources of measurement error. We randomized 64 participants to receive an AD or CS version of a scale measuring trust in medical researchers. We examine several indicators of data quality and cognitive response processing including: reliability, concurrent validity, recency, response latencies, and indicators of response processing difficulties (e.g., uncodable answers). Overall, results indicate reliability is higher for the AD scale, neither scale is more valid, and the CS scale is more susceptible to recency effects for certain questions. Results for response latencies and behavioral indicators provide evidence that the CS questions promote deeper processing. Qualitative analysis reveals five sources of difficulties with response processing that shed light on under-examined reasons why AD and CS questions can produce different results, with CS not always yielding higher measures of data quality than AD.