Chrome makes an attractive, durable finish, but it does have drawbacks. First, it involves many very hazardous chemicals, and the cost of dealing with these (safety and environment-wise) is reflected in the cost of plating. It is often considerably more expensive than paint. Second, good surface prep is essential to a good plating job. Unless you do the work yourself, you can expect this to cost at least as much as the plating itself. Finally, unless your plater is familiar with thin-wall tubing and complex joints, you may find the results inconsistent. The chemicals used in plating can corrode the tubing from the inside out, so good flushing is essential to protect the frame. And what flushes out needs to be treated as hazardous waste and disposed of appropriately.
Short version: there's a reason why chrome plating isn't common any more.
Are we talking about straight chroming, or chrome under paint?
As far as the drawbacks are concerned, it wouldn't make any difference; whether painted or not, the plating process and hazardous materials disposal problems are the same. Any chromed areas that are painted need not be quite as finished as those that won't be painted since every tiny flaw in the brazing and every file mark shows up like a sore thumb through naked chrome plating. OTOH, chrome plated areas that are to be painted have to have the surfaced roughed either chemically or mechanically so that the paint will have "tooth" to adhere to.
Because the chrome plating is a hard, abrasive resistant conversion layer, surface rust on plated surfaces is a non-issue until/unless it starts pitting. I have a fully chromed 1972 Schwinn Paramount that still looks brand new thanks to the excellent factory prep work and plating.
Done properly, chrome is a very tough and corrosion resistant finish. This means that if the paint is scratched it's no big deal.
I have several frames in the neighborhood of 30 years old. Because they are chromed under the paint -- a pretty common practice with top-of-the-line Italian racing frames of that time period -- they are still in perfect condition despite the normal chips and dings paint will acquire.
I also have several frames of the same age but without chrome under the paint. All of them have the cancer. So I guess you could say I'm a fan of chrome.