What exactly went wrong on regression coefficient p values for multicollinearity case? The following code shows a toy example for linear regression. I made almost identical columns and plug into linear model.
If we look at the significant code of the p value, I feel counter intuitive that if we use two significant variables but correlated, then both of them became not significant. 
what is the reason (the violated assumption) we have the wrong p value? Also the matrix condition number is high, is the numerical problem causing this?
> mtcars$wt2<-mtcars$wt+runif(32)*1e-5
> summary(lm(mpg~wt,mtcars))
Call:
lm(formula = mpg ~ wt, data = mtcars)
Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 
Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
wt           -5.3445     0.5591  -9.559 1.29e-10 ***


> summary(lm(mpg~wt+wt2,mtcars))
Call:
lm(formula = mpg ~ wt + wt2, data = mtcars)
Residuals:
    Min      1Q  Median      3Q     Max 
-5.4542 -2.0952 -0.2647  1.8365  6.3834 
Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)      38.71       2.20  17.593   <2e-16 ***
wt           206811.44  169901.52   1.217    0.233    
wt2         -206816.87  169901.60  -1.217    0.233    

 A: First of all the use of p-values in regression for significance testing  should be avoided. http://www.nature.com/news/scientific-method-statistical-errors-1.14700
Said that  let's discuss the problem from a strictly mathematical point of view. 
Given a dataset $(X, y)$ the least square estimator computes the projection of $y$ on the columns of $X$. If this projection on one of the columns is large enough and well defined then the p-value of the associated variable will be significant. 
In your second case you have two columns that are identical. As a consequence the cost function associate to the linear regression will probably be very flat for combinations of wt and wt2. 
I'm not sure but I would say that it has the same values for all the linear combinations for which $\alpha* wt + \beta *wt_2 \sim -5.34 *wt$
As a consequence the variance of the estimated coefficients explodes and make the p-value larger than 0.05. 
I am not 100% sure that all what I wrote is true but I am quiet confident it is. You can verify it numerically
