Can we cut and score on colored card stock paper?
As many have described in the comments, there are a variety of computer-guided cutting machines, ranging from the cricket and silhouette level machines which are designed for home use, to more serious home-manufacturing devices like the silver bullet.
These devices differ from each other in terms of price, the size of media they can work with and the variety of bits which are available for them.  They all share a very significant failing in the way that they secure the media to the cutting bed for the cutting/scoring process.  Each of them without exception (in my experience) use a sticky plastic mat to hold the media which is to be cut or scored.
That mat insures that the machine can maintain a firm grip on the media so that accurate registration can be maintained during the process.  But when dealing with certain media, for example card stock paper, that sticky plastic mat often does more harm than good.  The problem is that the adhesive on the mat is often stronger than whatever is holding the paper layers together.  Small cut pieces often tear and shred into wafers when after cutting, you attempt to remove them from the mat.
It is very frustrating to have a cut performed perfectly by one of these brilliant machines and then to unavoidably destroy the result while trying to release it from the adhesive mat.
There are low-adhesion mats available for some of these machines, but I have yet to find one which provides both enough holding power to prevent slippage during cutting/scoring and enough ease of release afterwards.