Training error and logistic regression

Suppose we have a training data set with variables $a,b$ and $c$ and binary outcome variable $y$. We fit a logistic regression model to this data set:

$$\text{logit}(p) = \hat{\beta_0}+ \hat{\beta_{1}}a + \hat{\beta_{2}}b + \hat{\beta_{3}}c$$

When we get the predicted probabilities from the training data set using the logistic regression model, are we supposed to get perfect classification? Or this depend on the threshold we use?

Assuming you prediction threshold is 0.5, if your training data are linearly separatable w.r.t. the independent variables (i.e., $a$, $b$, and $c$ in this case) and the optimization underlying the training process converges, yes, you can get a 0 training error.