This is more of comment, but I wanted to include a graph and some code.
I think the statement "if two predictors are correlated and both are included in a model, one will be insignificant" is false if you mean "only one."
Here's my counterexample using a regression of body fat percentage on thigh circumference, skinfold thickness, and midarm circumference:
. webuse bodyfat, clear
(Body Fat)
. reg bodyfat thigh triceps midarm
Source | SS df MS Number of obs = 20
-------------+------------------------------ F( 3, 16) = 21.52
Model | 396.984607 3 132.328202 Prob > F = 0.0000
Residual | 98.4049068 16 6.15030667 R-squared = 0.8014
-------------+------------------------------ Adj R-squared = 0.7641
Total | 495.389513 19 26.0731323 Root MSE = 2.48
------------------------------------------------------------------------------
bodyfat | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
thigh | -2.856842 2.582015 -1.11 0.285 -8.330468 2.616785
triceps | 4.334085 3.015511 1.44 0.170 -2.058512 10.72668
midarm | -2.186056 1.595499 -1.37 0.190 -5.568362 1.19625
_cons | 117.0844 99.78238 1.17 0.258 -94.44474 328.6136
------------------------------------------------------------------------------
. corr bodyfat thigh triceps midarm
(obs=20)
| bodyfat thigh triceps midarm
-------------+------------------------------------
bodyfat | 1.0000
thigh | 0.8781 1.0000
triceps | 0.8433 0.9238 1.0000
midarm | 0.1424 0.0847 0.4578 1.0000
. ellip thigh triceps, coefs plot(, yline(0, lcolor(gray)) xline(0, lcolor(gray)))
As you can see from the regression table, everything is insignificant.
The last Stata command graphs the confidence region for 2 of the regression coefficients (a 2 dimensional analog of the familiar confidence intervals). The confidence ellipse for the skinfold thickness and thigh circumference coefficients is long, narrow and tilted, reflecting the collinearity in the regressors. There's high negative covariance between the estimated coefficients. The ellipse covers parts of the vertical and the horizontal axes, which means that we cannot reject the individual hypotheses that the $\beta$s are zero, though we can reject the joint null that both are since the ellipse does not cover the origin. In other words, either thigh and triceps are relevant for body fat, but you can't determine which one is the culprit.