This is an intermittent problem, but more of a problem than not - hence the angst... After my mechanic installed a new cable, my right shifter (for rear) has not been working properly when I shift from my lowest gear (biggest cog in back) to my next gear up (second biggest cog in back). About 95% of the time, nothing happens when I push it (no click). I have to hold the brake lever out while I move the little lever in. This was happening while I climbed Mt. Washington last weekend - creating existential angst, if you know what I mean??? Other gears shift fine.
The other weird symptom: when it does shift, it has a delayed response meaning the click seems to happen AFTER I let go of the little lever. It aint normal!
Not having read "Shifters for Dummies", I've tried several long squirts of Pedro's dry lube and blasted copious amounts of compressed gas (the kind for dusting out computers) into the shifter - while in the highest cog (smallest) in back with quick release open and brake lever pressed. After doing this, occasionally, I can get a normal shift out of my lowest gear, but only a few times. This has not resolved my angst.
Not sure I'm skilled enough to go detaching the cable or re-installing one, but I suppose I will if I must...
Were new cable housings installed with the new cable? The housings are at least as important as the cable for good shifting performance. The short housing found just before the rear derailleur is often the sight for excess friction on the cable.
Also check the low limit screw adjustment. Use the instructions found at http://www.parktool.com for derailleur adjustments.
Edit: Also check up under the shifter for a possible loose or missing screw.