One of the OLS assumptions is "no perfect multicollinearity". So, if you don't have a perfect linear combination among your independent variables (IV), you do can estimate a OLS regression model.
Including a new IV that affect the DV and is uncorrelated with the other IVs is always good since it reduces error variance. However, if the new IV is somehow correlated with one or more IVs, the effect of increasing the error variance due to collinearity might be stronger.
One way to assess if the collinearity is harmful is evaluating the condition index. See example below:
x1 = rnorm(1000, 1, 1)
x2 = rnorm(1000, 2, 1)
x3 = rnorm(1000, 3, 1)
x4 = 3*x3 + rnorm(1000, 1, 0.1)
library(perturb) # colldiag
colldiag(cbind(x1, x2, x3, x4))
## Index Variance Decomposition Proportions
## intercept x1 x2 x3 x4
## 1 1.000 0.000 0.017 0.009 0.000 0.000
## 2 2.933 0.000 0.928 0.015 0.000 0.000
## 3 4.766 0.000 0.000 0.755 0.000 0.000
## 4 9.129 0.129 0.051 0.216 0.000 0.000
## 5 289.166 0.870 0.004 0.006 1.000 1.000
Belsley, Kuh, and Welsch (1980) say:
It is suggested that an appropriate means for diagnosing degrading collinearity is the following double condition:
A singular value judged to have a high condition index, and which is
High variance-decomposition proportions for two or more estimated associated with regression coefficient variances.
They suggest a condition index greater then 30 with variance-decomposition proportions greater than 0.5.
In the example above, you have a condition index of 289.166 associated with two variables (x3 and x4) with variance-decomposition proportion of 1. In this case, it would be advisable to exclude one of these variables from the model.