Why is it possible to get significant F statistic (p<.001) but non-significant regressor t-tests? In a multiple linear regression, why is it possible to have a highly significant F statistic (p<.001) but have very high p-values on all the regressor's t tests?
In my model, there are 10 regressors. One has a p-value of 0.1 and the rest are above 0.9

For dealing with this problem see the follow-up question.
 A: The answer you get depends on the question you ask.  In addition to the points already made, the individual parameters F values and the overall model F values answer different questions, so they get different answers. I have seen this happen even when the individual F values are not that close to significant, especially if the model has more than 2 or 3 IVs.  I do not know of any way to combine the individual p-values and get anything meaningful, althought there may be a way.
A: One other thing to keep in mind is that the tests on the individual coefficients each assume that all of the other predictors are in the model. In other words each predictor is not significant as long as all of the other predictors are in the model. There must be some interaction or interdependence between two or more of your predictors.
As someone else asked above - how did you diagnose a lack of multicollinearity?
A: As Rob mentions, this occurs when you have highly correlated variables. The standard example I use is predicting weight from shoe size. You can predict weight equally well with the right or left shoe size. But together it doesn't work out.
Brief simulation example
RSS = 3:10 #Right shoe size
LSS = rnorm(RSS, RSS, 0.1) #Left shoe size - similar to RSS
cor(LSS, RSS) #correlation ~ 0.99

weights = 120 + rnorm(RSS, 10*RSS, 10)

##Fit a joint model
m = lm(weights ~ LSS + RSS)

##F-value is very small, but neither LSS or RSS are significant
summary(m)

##Fitting RSS or LSS separately gives a significant result. 
summary(lm(weights ~ LSS))

A: Multicollinearity


*

*As you note, and as has been discussed in this previous question, high levels of multicollinearity is one major cause of a statistically significant $R^2$ but  statically non-significant predictors.

*Of course, multicollinearity is not just about an absolute threshold. Standard errors on regression coefficients will increase as intercorrelations with the focal predictor increase.


Multiple almost significant predictors


*

*Even if you had no multicollinearity, you can still get non-significant predictors and an overall significant model if two or more individual predictors are close to significant and thus collectively, the overall prediction passes the threshold of statistical significance. For example, using an alpha of .05, if you had two predictors with p-values of .06, and .07, then I wouldn't be surprised if the overall model had a p<.05.

A: This happens when the predictors are highly correlated. Imagine a situation where there are only two predictors with very high correlation. Individually, they both also correlate closely with the response variable. Consequently, the F-test has a low p-value (it is saying that the predictors together are highly significant in explaining the variation in the response variable). But the t-test for each predictor has a high p-value because after allowing for the effect of the other predictor there is not much left to explain.
A: One way to understand this is the geometry of least squares as @StasK suggests. 
Another is to realize it means that X is related to Y when controlling for the other variables, but not alone. You say X relates to unique variance in Y. This is right. The unique variance in Y, though, is different from the total variance. So, what variance are the other variables removing?
It would help if you could tell us your variables.
A: Consider the following model: $ X_1 \sim N(0,1)$, $X_2 = a X_1 + \delta$, $Y = bX_1 + cX_2 + \epsilon$, $\delta$, $\epsilon$ and $X_1$ are all mutually independent $N(0,1)$.
Then $${\rm Cov}(X_2,Y) = {\rm E}[(aX_1+\delta)(bX_1+cX_2+\epsilon)]={\rm E}[(aX_1+\delta)(\{b+ac\}X_1+c\delta+\epsilon)]=a(b+ac)+c$$
We can set this to zero with say $a=1$, $b=2$ and $c=-1$. Yet all the relations will obviously be there and easily detectable with regression analysis.
You said that you understand the issue of variables being correlated and regression being insignificant better; it probably means that you have been conditioned by frequent mentioning of multicollinearity, but you would need to boost your understanding of the geometry of least squares.
A: A keyword to search for would be "collinearity" or "multicollinearity". This can be detected using diagnostics like Variance Inflation Factors (VIFs) or methods as described inthe textbook "Regression Diagnostics: Identifying Influential Data and Sources of Collinearity" by Belsley, Kuh and Welsch. VIFs are much easier to understand, but they can't deal with collinearity involving the intercept (i.e., predictors that are almost constant by themselves or in a linear combination) - conversely, the BKW diagnostics are far less intuitive but can deal with collinearity involving the intercept.
