If I have many positive, insignificant results, can I test "at least $n$ of these results are positive"? Let's say I have run the same regression for 100 different individuals separately. My coefficients of interest are positive (and quite different from one another) but statistically insignificant in all 100 results (lets say every p-value = 0.11).
Is there a way to combine these p-values in order to conclude "at least 80 of these results are positive" with greater significance than p=0.11? My online searches have only showed me how to say "at least 1 of these results is positive" through a Fisher or similar test, but I haven't been able to generalize that result. I want to test "H0 = all 100 effects are the same at 0" against "HA = at least 80 effects are positive."
My goal isn't to say there is a positive coefficient on average, and nor is it to specifically measure the coefficient. My goal is to demonstrate, with significance, that at least 80 people individually faced some positive effect regardless of which 80, and regardless of the magnitude of effect felt by each individual.  
 A: You should perform all 100 analyses as a single mixed effects model, with your coefficients of interest random variables themselves.  That way you can estimate a distribution for those coefficients including their overall mean, which will give you the sort of interpretation I think you are looking for.
Noting that, if as I suspect is the case, you have a time series for each individual, you will also need to correct for autocorrelation of the residuals.  
A: The simplest thing to do would probably be a sign test.  The null hypothesis is that each result has equal probability of being positive or negative (like flipping a fair coin).  Your goal is to determine whether the observed results would be unlikely enough under this null hypothesis that you can reject it.
What's the probability of getting 80 or more heads out of 100 flips of a fair coin? You can calculate this using the binomial distribution. In R, the relevant function is called pbinom, and you could get a (one-sided) p-value using the following line of code:
pbinom(80, size = 100, prob = 0.5, lower.tail = FALSE)
According to this test, your intuition is correct, you'd be exceedingly unlikely to get 80 positive results by chance if the treatment had no effect.
A closely related option would be to use something like the Wilcoxon signed rank test.

A better approach, if you actually want to estimate the size of the effect (rather than just determine whether it tends to be greater than zero or not), would probably be a hierarchical ("mixed") model.
Here, the model says that your 100 individuals' results come from a distribution, and your goal is to see where the mean of that distribution is (along with confidence intervals).  
Mixed models let you say quite a bit more about your effect sizes: after fitting the model, you could say something like "we estimate that our treatment tends to improve outcomes by an average of three units, although the data is consistent with the true average effect size being anywhere from 1.5 to 4.5 units. Also, there's some variation among individuals, so a given person might see an effect anywhere from -0.5 to +6.5 units". 
That's a very precise and useful set of statements--much better than just "the effect is probably positive, on average", which is why this approach tends to be favored by statisticians.  But if you don't need all that detail, the first approach I mentioned could be fine too.
A: Maybe I get that completely wrong, but what it seems to me is that you are trying to do repeated-measures ANOVA. Just define this "dummy" as a within-subject factor, and the model would do the rest. Significance itself is not very informative; it is required but not sufficient; any model would get significant with a sufficiently large number of observations. you may want to get effects size, like (partial) Eta-Squared, to get an idea of how "big" your effect is. 
My 2 cents.
A: It might be as simple as an ordinary ANCOVA calculation, but the appropriate way to analyze your data would depend on the physical situation and you haven't supplied those details.
