Making up for a low sample size by increasing the number of observations What exactly is the error that is made in the following judgement?
"I cannot increase my sample size, so I'll make up for the low number of subjects by asking the subjects to perform more trials, i.e. by increasing the total number of observations / data points" ?
Does an explanation based on degrees of freedom and the repeated measures vs between-subjects nature of the data collection explain this error of judgement?
 A: Simple counter-example would be probably enough for you to figure the answer yourself. Say, that you have to make a study on human height, however you do not have enough time to conduct a full-scale survey. In this case you decide to ask your roommate 1000 times about his height.
A: Asking the same person to perform the same trial will increase your information about within-person variance but not between-person variance. If within-person variance is what you're interested in, then the logic makes sense. If it's not what you're interested in, then it doesn't.
A: Another way to look at this is to consider that all the measures you take reflect a systematic source of variance (which you are interested in) compounded by various sources of errors (i.e. variance you are not interested in).
Depending on your objectives and the relative magnitude of these sources of errors, different designs might make sense but you can't learn about or compensate one source of error (e.g. individual differences between participants) by increasing the number of observations in another facet of the design (e.g. repeated observations of the same participant over time).
Generalizability theory is a way to formalize this insight (and the source of the terminology I used in this answer).
