What IRstuff said is true. It is difficult for most people to see any type of water as a corrosive substance, but DI water can be extremely corrosive in the right environment.
I belive quark is talking about a situation that is different from the one you are experiencing. It sounds like you are hardly using the DI system, and sometimes the conductivity reading go up for, seemingly, no reason. This same thing happens in one my plants because we installed a DI system for a process that is no longer in use. The water is now only used for lab activities, so the system stays stagnant for weeks at a time. If that is the case, you may also find that the water "stinks" due to organic growth in the piping system. Running the system for a few minutes normally fixes both problems.
If you try to measure the pH of DI water, you will notice that the value returned is not the expected 7.0, but, typically down around 4.0. The reason is that the DI water dissolved CO2 from the air which forms carbonic acid. Actually, this has as much to do with the measurement itself than anything. The instrument needs something to read, and the very small amount of acid in the water is enough to give a false low pH. A stabilizer (KCl, I think?) needs to be used to measure an acurate pH in this case.