This is a common misconception. The Vce(sat) of most common transistors is less than the bias voltage required on the base to create that saturation.If via bipolar transistor there is a voltage drop of about .7V that the connection will remain above actual ground.
We're getting off topic here (since PICAXE outputs aren't fitted with internal TIP31s) but you left out an important parameter:I concur with IWP.
For small signal transistors such as the BC547/548/549 when used for signal purposes, I usually use a value of 0.2 Volts for Vce(sat).
Certainly as the Collector current (and required base current) draw increases then Vce(sat) also increases. For example, with a 3 Amp TIP31 transistor at Ic = 3 Amps and Ib=0.375 Amps, the Vce(sat) can be as high as 1.2 Volts.