Recent psychophysical studies suggest that normal-hearing (NH) listeners can use acoustic temporal-fine-structure (TFS) cues for accurately discriminating shifts in the fundamental frequency (F0) of complex tones, or equal shifts in all component frequencies, even when the components are peripherally unresolved. The present study quantified both envelope (ENV) and TFS cues in single auditory-nerve (AN) fiber responses (henceforth referred to as neural ENV and TFS cues) from NH chinchillas in response to harmonic and inharmonic complex tones similar to those used in recent psychophysical studies. The lowest component in the tone complex (i.e., harmonic rank N) was systematically varied from 2 to 20 to produce various resolvability conditions in chinchillas (partially resolved to completely unresolved). Neural responses to different pairs of TEST (F0 or frequency shifted) and standard or reference (REF) stimuli were used to compute shuffled cross-correlograms, from which cross-correlation coefficients representing the degree of similarity between responses were derived separately for TFS and ENV. For a given F0 shift, the dissimilarity (TEST vs. REF) was greater for neural TFS than ENV. However, this difference was stimulus-based; the sensitivities of the neural TFS and ENV metrics were equivalent for equal absolute shifts of their relevant frequencies (center component and F0, respectively). For the F0-discrimination task, both ENV and TFS cues were available and could in principle be used for task performance. However, in contrast to human performance, neural TFS cues quantified with our cross-correlation coefficients were unaffected by phase randomization, suggesting that F0 discrimination for unresolved harmonics does not depend solely on TFS cues. For the frequency-shift (harmonic-versus-inharmonic) discrimination task, neural ENV cues were not available. Neural TFS cues were available and could in principle support performance in this task; however, in contrast to human-listeners' performance, these TFS cues showed no dependence on N. We conclude that while AN-fiber responses contain TFS-related cues, which can in principle be used to discriminate changes in F0 or equal shifts in component frequencies of peripherally unresolved harmonics, performance in these two psychophysical tasks appears to be limited by other factors (e.g., central processing noise).