The first thing that comes to my mind when I read of the approach that you describe is that there is a miss-match between the idea of replicating an experiment and the use of "success" and "failure" as descriptors of the outcomes. Presumably a success would be a result that is significant in the Neyman-Pearson paradigm (i.e. P < alpha). However, that paradigm allows control of type I errors and specification of power by assuming that the experimenter will act as if the null hypothesis is false when a significant result is observed. In that case there is no point in testing it again--the determination of type I error rates assumes that you discard the hypothesis and move on. Neyman called the process 'inductive behavior'.
A consequence of inductive behavior is that you can't integrate the information across multiple replications of the overall experiment within the Neyman-Pearson paradigm. There would be no sensible interpretation of any sequence of significant and not significant results from repeats of the experiment using the Neyman-Pearson paradigm.
If you use instead Fisher's approach of inductive inference, then you can combine P values from multiple experiments testing the same null hypothesis and obtain a composite value. The P values are indices of the strength of evidence against the null hypothesis (and NOT error rates!) and so, just as it is sensible to combine the evidence from multiple sources before judgement, you should combine the P values. However, while there are several (many?) way to combine the P values, it is perhaps the case that none is perfect.
Arguably the best way to combine evidence from multiple experiments is to use a likelihood-based approach (see Royall 1997 for a very clear exposition) . The likelihood functions for each of the experiments can be combined exactly by multiplication to yield a composite function that can support both interval estimation and statement of the probability of the overall set of results having come from a true null hypothesis.
Royall, R. M. (1997). Statistical evidence: a likelihood paradigm. Chapman & Hall.
It is unfortunate that Fisher was frequently unclear in his expression, and rather inconsistent in his application of the ideas. However, he was undoubtedly in favor of repeating experiments and hated Neyman's idea of inductive behavior:
“...we may be able to validly apply a test of significance to discredit a hypothesis the expectations from which are widely at variance with the ascertained fact. If we use the term rejection for our attitude to such a hypothesis, it would be clearly understood that no irreversible decision has been taken; that, as rational beings, we are prepared to be convinced by future evidence that appearances were deceptive, and that a remarkable and exceptional coincidence had taken place.” Stat Meth & Sci Inf p.37 (Chapter II section 4).