Correlation not significant because there is not enough variance? I have a question about correlations again. 
I have a dichotomous variable that I want to correlate with the another one (metric) by using the point-biserial correlation coefficient. I get a non-significant result (n=28). But my concern is that one of the variable has almost no variation in the values: 27 observations are $1$, 1 observation is $0$! 
In other words, no matter what value variable 2 has, variable 1 has in almost every case the value 1.
Can this be the reason why a correlation is not significant?
 A: It's possible to get high point biserial correlation even with 27 $1$'s and a $0$. Indeed, you can get as high as 1, so it's not that:
  y <- c(0,rep(1,27))
  x <- y
  cor(x,y)
[1] 1

-- and making x continuously distributed doesn't substantively alter that conclusion:
  y <- c(0,rep(1,27))
  x=c(rnorm(1,0),rnorm(27,100))
  cor(x,y)
[1] 0.9987537

This corresponds to a $t$ statistic of 102.036 on 26 d.f.
On the other hand, see the advice in the wikipedia page on the point-biserial:

One disadvantage of the point biserial coefficient is that the further the distribution of Y is from 50/50, the more constrained will be the range of values which the coefficient can take. If X can be assumed to be normally distributed, a better descriptive index is given by the biserial coefficient

Since we saw it's possible to get a high correlation, you might wonder what the restriction can be. I assume that they mean "if we restrict $x$ to be normally distributed" -- because then it can be limited:
 x <- sort(rnorm(28,100))
 (rpb <- cor(x,y))
[1] 0.4682661

Much smaller. But still potentially significant:
 pt(-rpb*sqrt(26/(1-rpb^2)),26)*2
[1] 0.01196686

A thousand simulations suggest that when the smallest x goes with the lone zero, the correlation is mostly between 0.3 and 0.5, and about 60% of the values are significant at the 5% level.
So if the continuous variable is restricted to have a normal margin, even then the correlation - while typically a lot smaller than what it otherwise possible, can still be significant.
A: @Glen_b examples showed that a strong correlation is not strictly impossible in this situation but I think your intuition is right. Limited variation (for example range restriction) can bias sample correlations toward 0. The problem is not merely one of “significance”, the correlation also appears systematically smaller than it would be if you would consider the whole population.
28 is not a huge sample either so perhaps the problem is simply a lack of power.
