I performed a logistic regression of an outcome variable (whether a patient is re-admitted to a hospital within a year) against a continuous index that measures a patient's access to healthcare. I used the glm function in R with family = 'binomial' for the analysis.
I obtained a small negative logit regression coefficient with p = 0.54 (based on a standard normal distribution for the coefficient beta/se(beta) ~ z). In spite of the fact that the regression coefficient was not significantly different from 0, the McFadden pseudo-R-squared value = 0.63, suggesting that the likelihood associated with the regression model is much higher than that of the null model, in spite of the regression coefficient not being significantly different from the latter.
If I were performing a multivariable regression with a large number of predictor variables, it is certainly possible for some (or all) of them not to have statistically significant regression coefficients while having an overall model with a high (pseudo)R-squared value. I do not understand how this would happen with a single predictor variable, either in this context (logistic regression and pseudo r-squared), nor indeed in a simple linear regression model with a single predictor.
Scatterplots of the data are consistent with a poor fit and a non-significant p-value, so where the pseudo r-squared approaching 1 is coming from is a mystery to me.