Ice bath is always 3C, why? I've always learned that a mixture of ice and water should reach equilibrium at approximately 0C.  I've actually tried to create that a number of times in different contexts and always fail.
First, an ice bath in a plastic container.  I get 3C, occasionally as low as 2C with a multimeter thermocouple.  Granted those are inaccurate, so I've tried a digital thermometer from Vernier with claimed accuracy of 0.5C.  I confirmed with an IR thermometer. Same story.
Thinking that perhaps it is due to warming from the outside, I have put the ice bath in a thermos and chilled it a while.  Same story.  I am not using distilled water, so there are some minerals in the water, but if anything I thought that would lower the temperature of the ice bath.
 A: Hypothesis
Ideally, the ice and water should reach an equilibrium at zero celsius. But this equilibrium might take a long time to happen, based on the exact setup.
Looking at a typical setup of yours, the ice will float at the top of the container and there is water (but no ice) at the bottom. Water is densest at $\sim 4$ degrees celsius, and such water will sink to the bottom of the container. The water at the top is in contact with the ice and so, it must be near zero celsius, or getting cooler. 
So I imagine a situation where we have ice and cold water on top, near the surface (which has to be in equilibrium with the atmosphere, but let us neglect that for now). There is also cold water at the bottom of the vessel (at 4 degrees celsius or cooler), with a temperature gradient leading to the top. The actual temperature profile will depend on all kinds of non-equilibrium physics having to do with the shape/size/conductivity of the container, the size of the ice cubes, etc.
My guess is that since you would have submerged the thermometer a significant depth into the container (maybe close to the bottom), you're measuring the temperature to be closer to 4 degrees celsius than to zero celsius.
Test
If my hypothesis is correct, then by stirring the contents of the container, you should be able to set up convection currents which will cool the contents more uniformly. You might want to be careful in not stirring the container too vigorously, for that will heat up the contents. I think gentle stirring should do the job without infusing much heat into the system, at least at the level of accuracy of your measurements.
A: The calibration of the digital devices drifts. 
When calibrating some temperature probes for a neutrino experiment we used a deionized-water ice bath. The four laboratory digital thermometers we found (all claiming between $\pm 0.05$--$0.25\,^\circ \mathrm{C}$ accuracy) read between $-0.5$ and $+1.8\,^\circ\mathrm{C}$. Clearly some were well outside their claimed uncertainty.
We then went down to the chemistry stockroom and bought a freshly calibrated device which read $0.05 \,^\circ\mathrm{C}$. 
An antique, 18 inch mercury expansion instrument we turned up the next day read $32.00 \pm 0.05 \,^\circ\mathrm{F}$. There is something to be said for the old ways.
One lesson here is that you have to be careful about the trustworthiness of instruments over time.
