Does the speed of an ion affect the current that can flow through a solution? Typically, when a cathode and anode are inserted into a solution that allows current to flow, the ions are moving with a certain kinetic energy.

If the ions were to gain kinetic energy (through transfer of thermal energy for example), will more current be able to flow, and will the potential difference increase?

My assumption for this is that increased ion kinetic energy will speed up the rate of ions being attracted to each electrode, thus causing more charge to arrive in a given period of time. As $I = Q/t$ where $I$ = current, $Q$ = charge and $t$ = time, would this mean an increased current?

Edit: I've performed a basic experiment in which I got a Tupperware container, stuck in some multimeter probes, and tested the difference in potential (voltage) for $\ce{NaCl}$ (table salt) in solution when cold and hot, then for water cold and hot. I appreciate there are many issues with reliability, but the general results were:

Water cold = 0.082 V
Water hot = 0.000 V (actually went into the negative then stabilised at 0)
NaCl cold = 0.065 V
NaCl hot = 0.130 V  

Now, my real experiment involved $\ce{AgNO3}$ being titrated into $\ce{KCl}$ (mixed with deionised water and a little $\ce{HNO3}$ to acidify) – assuming the temperature remained constant, what effect could increasing the ions' velocity have on conductivity, without heating it up (increasing kinetic energy) to alter viscosity?
There is a strong dependence on temperature in conductivity of ionic solutions; see http://www.eutechinst.com/techtips/tech-tips44.htm, for example. However, thermal energy, i.e. molecular motion, is random, and does not favor current flow in the forward (or backward) direction. There are a number of factors that temperature influences:


Viscosity usually changes  inversely with temperature, so ions move more freely at higher temperature.
Ionic solutions have other species than OH- and H+; for example OH3+ [water + hydroxyl radical] is the hydronium ion; see http://en.wikipedia.org/wiki/Hydronium. The amount of each species varies with temperature, as shown in that reference. See also http://en.wikipedia.org/wiki/Self-ionization_of_water. Depending on mass, hydrogen bonding etc. different species have different conductivity.