The number on the box tells you what colour the light is. It does not tell you how bright it is, and it does not tell you whether the things under it will look like themselves — which is the part most people actually care about.
Heat a theoretical perfect radiator and it glows: dull red first, then orange, then white, then blue-white. Correlated colour temperature names the point on that progression a lamp most closely resembles. The apparent contradiction — that higher numbers look cooler — is because the scale describes physical temperature while the words warm and cool describe how the light feels. Nobody has ever successfully renamed them.
The important limitation is that it is a summary of one dimension. Two lamps can share a rating and distribute their energy across the spectrum quite differently, and the objects in a room can only reflect wavelengths that are actually present. That is why colour temperature is necessary information and never sufficient information.
Approximate on purpose. The boundaries are conventions and vendors place them differently; what is reliable is the ordering.
around 2200K
around 2700K
around 3000K
around 3500–4000K
around 4000–5000K
5000K and above
Colour temperature and colour rendering are independent, and only one of them is usually on the box. Temperature decides what the light looks like; rendering decides whether the things under it look like themselves. A warm lamp that renders well is a pleasant room; a warm lamp that renders badly is the reason a grey wall turns green after dark. If a space involves judging colour — hair, makeup, fabric, food, paint — rendering is the specification that matters and temperature is a matter of taste.
Colour rendering, and why it matters more than temperature