What to do with Categorical variables that are partially significant? Suppose you have a very simple linear model to predict Salary.  Your ridiculously simple model is based on a persons age and their education level which can have the values: None, High School, College, Post Grad.
gm0 <- lm( Salary ~ Age+EducationLevel, data=dfWork )
summary(gm0)

Now you run the summary.  The coefficient for Age turns out to be so significant that p is almost zero.  As for education level the results are mixed.  The only value with a significant p is EducationLevel=College.
So ... umm ... what do you do with this information?  Do you include EducationLevel in the model even though only one of the values is significant?  Do you scrap it because only one value is significant?  Or do you make a new column with a value=1 when EducationLevel=College, 0 otherwise and include THAT categorical variable into the model (and not the original EducationLevel).
 A: I would not do anything on the basis of "statistical significance" alone.
First of all, the choice of the significance level is completely arbitrary and often dictated by ritual. 
Second, "statistical significance" is highly dependent on sample size.
Third, "statistical significance" says absolutely nothing about practical significance. You might have a highly statistically significant estimate of an extremely small and uninteresting effect.
Fourth, the non-statistically significant estimate may be interesting in itself.
Fifth, re-categorizing a variable in order to chase p-values could completely mislead your audience, if the removed levels do have an association with the outcome but you simply didn't have enough statistical power.
Sixth. Form a research hypothesis (perhaps consider that there might also be an interaction and/or non-linear association), conduct a power analysis, collect the data, fit the model corresponding to your research hypothesis, check model assumptions, and report the unadulterated results.
A: You should ignore what you've already done and run a subset (joint) F-test of significance for the entire educational variable (all dummy variables for edu). If the p-value for this subset F-test is significant, then go ahead and look at the individual tests for the different levels of education in reference to the base group. Irrespective of p-value for the individual levels of education, you should leave them all in (as a general rule) because the subset F-test told you the variable of "Education" was important and it's nonsensical to remove portions of this variable and can turn into p-hacking. If the subset F-test is not significant, then do not make the individual comparisons of educational level. There are some very good points made by @Robert Long.  
A: If only one of the factors in EducationLevel is significant consider omitting EducationLevel as a whole. If you want to keep EducationLevel but with only one or two factors, consider collapsing those factors to only include rows in your data that contain the desired factor levels. A way to do this:
library(tidyverse)

gm0 <- gm0 %>%
  filter(EducationLevel==College)

However, modeling is an art form, there is no correct answer. You may keep all EducationLevel factors even if certain factors aren't significant if the entire model is significant. That is up to you.
