Weldable primer is very commonplace for prefabricated pipe and structural members. It is an excellent rust prohibiting coating and far easier than any other paint to remove for the poor welders in the field. Theory behind it is that the aluminum (often added to filler metals and fluxes) is a deoxidizing agent. However, the binding/coating agents of the paint itself is still undesirable to weld through if applied too heavily. Oh yeah, DO NOT attempt to GTAW through this stuff. Unless you enjoy prepping and changing out tungsten.
Worst I've had to remove was bearing grease that the painters applied when told to protect the prepped ends from paint. Even if you get it "all" off from the weld zone, the heat after welding begins will cause it to leach in and stink to high heaven.

I don't believe in weldable paint.  I know it can be done but it's just cutting corners and isn't the way to do things.

A couple of years back I was working in a steel building and there was a rumour going around that you could weld without removing the paint.  I don't know who started it.  When the inspector came up to take a look at things I told him.  His response - "Bull $hit"!  - and I laughed and told him he might get an argument from some of the guys. 
He laughed and said, "You know, arguing with an inspector like me is a bit like wrestling with a pig in a mud hole - you soon get the idea that the pig is enjoying himself."

Besides, I don't like the paint fumes.  You get enough of them as it is.
Tom

As a CWI, I know what the codes say about welding on anything but clean metal and understand why.
It is simply to ensure the best possible result, defect reduction, and protect against under bead cracking from hydrogen embrittlement.
As a former welder in the real world, I also know that not just primer, but many different types of coatings are welded through every day with good result.
If a company is going the extra mile to pay extra for "weldable primer", my hat is off too them for doing more than many.
There are a lot of oil rigs and ships out there that are covered with welds that were made according to the following "procedure":
First, get some 6010 or 6013 rods if you've got them.
Next chip a hole through however many coats of epoxy based paints and primers there are on the steel to be welded. Note: don't waste time chipping off all the paint in the weld area, just chip a hole down to bare metal so you can strike an arc.
Next, tack your pieces together at the chipped location.
Then, if you've got one handy, fire up your cutting torch and burn the paint along the joint. Hit the oxygen to blow away as much paint ash as you can, then wire brush.
Make a root pass with the 6010 or 13 in order to finish burning the paint.
Cap with 7018 or FCAW.

I'm not saying it's right, or that I would accept it if inspecting. I'm just saying that it happens all the time because it works.

Tim