Precision of probabilities in lm or glm

I have major experimental data which appear suitably linear with the 
diagnostic plots of lm or glm. The probabilities for coefficients for over a 
dozen variables of low inter-correlation are of the order of 10e-6 to 10e-12. 
Analyis of trimmed data to exclude outliers increases that range to 10e-16. 
Such extreme data lead me to look for every possible explanation. Can anyone 
tell me the precision of these results for lm or glm. Am I dealing with 
numbers that are meaningless beyond a certain point or are there other things 
I should be examining in the data?

Gordon M. Harrington            Mail:   3720 Village Place, #6308
Professor Emeritus                      Waterloo, IA 50702-5848
University of Northern Iowa     Phone:  319-291-8535
gordon.harrington at uni.edu       Fax:    319-291-8491
dryfly at aya.yale.edu                     319-291-8324