Below is some example of data to aid the conversation rather than speaking about 'random-effects meta-analytic linear models' in which case I do not really know what you want to do. I hope the example may help you and maybe you can add some more specific example of your own to place your problem in a better perspective.
You have two aspects to deal with.
- can we make predictions based on few data points?
- hypothesis test accept/reject
The example below illustrates the first aspect.
FEW DATA POINTS
By, using the assumption that the variance remains equal you can make estimates of confidence intervals even if you have only a single measurement for some level. This is because you can use the information of the many measurements at the other levels to make an estimate of the variance of the measurements. Say, the variance is estimated to be about x and you find a point at a distance 2 times x from some boundary, then you can consider the point as significant.
Take note of the the two very close measurements at 3 on the x axis in the image below. Some people would create an estimate of the standard error of the means, separately for each of the levels, and then decide that the points that happen to be very close, relate to an accurate estimate of the mean for this level with a low standard error. (this could happen by chance, and this example which is randomly generated data, demonstrates the difficulty in such reasoning)
HYPOTHESIS REJECTION/ACCEPTION
On this aspect you can go into many directions with technical details. And there is no clearcut answer.
I do not entirely get your comment about
"If all of them just would have come out non-significant, I wouldn't really have to worry about them that much, but some of them come out significant."
1) What does it mean 'to come out non-significant'?
Your zero faith means that you suspect false positives?
One think that you have to look into is how to correct for, or deal with, multiple comparisons, and you can use many options which depends a bit on your situation and preference.
2) I also wonder what you are doing with your results. What are the practical consequences of significant results?
Instead of worrying about false positives, I would worry about false negatives. If you only have a small amount of measurement points then the error in your estimate, or the spread of the posteriori probability distribution (whatever flavour of statistics interpretation you like), will be large. This means that you would easily accept the null-hypothesis, although your test is too weak in order to detect the effects.
You should also consider whether some relevant limit for the effect size isn't inside the boundaries of the range of the estimated effect-size or in the range of the posteriori probability distribution. You might say that the outcome is non-significant if the null-hypothesis is inside the 95% confidence interval... but what if some relevant effect size is also within the confidence interval (because that interval is so big).
This, and variations of the theme, you can use to determine how many measurements are needed or how to interpret results that are neither significant in the sense of tests for the null-hypothesis, nor strong enough to detect the lowest boundary for a relevant effect size.
x <- c(rep(1,20),rep(2,3),rep(3,2),rep(5,11),rep(6,1),rep(7,4))
y <- 4 * x + qnorm(runif(length(x),0,1),0,2)
data <- list (x=x,y=y)
m<-lm(y~factor(x),data)
var<-predict(m,interval="confidence")
plot(x,predict(m),ylim=range(var),pch=21,col=1,bg=1)
arrows(x, var[,2], x, var[,3], length=0.05, angle=90, code=3)
points(x,y,ylim=range(var),pch=1,cex=0.8)
