regression coefficient sign is opposite of correlation but VIF is low? I have used best subset regression with various selection criteria (e.g. Mallow's Cp and AIC) to compile a list of possible regression models to explain the response variable "Length". Below is the best model of the bunch,
Length = 5.770 + 0.1180 Max flow field width + 0.02691 time until ff + 0.1897 Duration of FF - 0.0465 Cooling dominate phase
however the coefficient for "cooling dominate phase" is negative in the regression but it has a positive correlation to the response variable. Here is the table correlation coefficients for the response and predictor variables in the model.

The VIF scores are all under 2 but being above 1 it suggests there is some degree of collinearity or multicollinearity happening. Here are the VIF's. 

Can I still have confidence in the accuracy of the regression coefficients if the VIF's are low but the sign of a predictor variable coefficient is opposite from its correlation to the response variable?  
 A: It's possible that the regression coefficients from a one-predictor regression and from a multiple-predictor regression being different. The signs do not need to agree, so aren't their magnitudes. So, what you see here is not abnormal.
While the VIF are not "high," the individual $R^2$ are not small. For example, take VIF 1.94:
$$VIF = \frac{1}{1 - R^2_{duration\ of\ FF}}$$
$$1/1.94= 1 - R^2_{duration\ of\ FF}$$
$$ R^2_{duration\ of\ FF}= 1 - 1/1.94$$
$$ R^2_{duration\ of\ FF}= 0.484$$
About 48% of the variability in "Duration of FF" can be explained by the other three independent predictors. Their correlations are not as weak as you think.

Can I still have confidence in the accuracy of the regression
  coefficients if the VIF's are low but the sign of a predictor variable
  coefficient is opposite from its correlation to the response variable?

Since you mentioned you selected many subsets so I guess your doing some sort of predictive model? If so, does this version give you the best prediction?
