I'm not sure I understand, but in summary:

i) if you have a conventional covariate that is measured once for each person/unit then simple ANCOVA is not helpful because the covariate simply partials out variability that is part of the subjects term (which does not influence the denominator of the F ratio used to test effects)

ii) in a more complex model that includes interactions with covariates such covariates are potentially useful or interesting and can influence inferences and tests

iii) if you have a time varying covariate (one that takes different values over the repeated measures) then this is of interest in all models, but can't be fitted with conventional repeated measures ANOVA. This is the sort of situation where a multilevel (or linear mixed model) is a sensible way to extend the ANOVA type approach.

I think “interdependent” might mean that for a within-participants design the assumption of independence of errors is not given, because there is more than one data point per participant.

Regarding requirements: An ANCOVA has the same requirements as an ANOVA and some additional requirements. A good overview about the additional requirements is given in the following paper:

Mulligan, N. W., & Wiesen, C. (2003). Using the analysis of covariance to increase the power of priming experiments. Canadian Journal Of Experimental Psychology/Revue Canadienne De Psychologie Expérimentale, 57(3), 152-166. doi:10.1037/h0087422

Hi Catherine, This reaction comes probably too late. But the other day I came across a very helpful article concerning ANCOVA of designs with only within-ss factors or with both within-ss and between-ss factors: http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00474/full.

I am afraid that I perhaps misunderstand you. In that case my following remarks are probably superfluous. Not being interdependent means that the scores across the time points don't depend on each other, i.e. are not correlated. This is known as the assumption of sphericity. But very often persons tend to have approximately the same scores on each time point. In that case the assumption of sphericity is violated. The Mauchly test, standard in SPSS output in case of (more than 2) repeated measures, tests whether this assumption is violated. If it is violated, the multivariate F-tests in the output of the ANOVA should be used or one of the corrected Fs (such as Greenhouse-Geisser) in the table of within-ss effects. (In case of just two measurement times, there is no problem of sphericity because there is only 1 score per individual, i.e. the difference between the two times.)

I don't know of a possible special assumption in case of repeated measures and an ANCOVA. Of course, besides the assumption of sphericity for the repeated measures part of the design, there is also the important specific ANCOVA assumption of equal regression slopes (or regression coefficients) across groups aka zero between-ss factor x covariate interaction. But this only applies when the design also contains between-ss factors.

Thank you Ernest, your answer is still very helpful