I was stumped but the person who told me about the issue figured it out.
It is because not all fonts have a bold alternative available.
To see this, open the Labelling Properties for any layer, and go to Appearance under the Symbol tab. If you look at Tahoma (which is a font to which I often apply bold text formatting) you can see that it has Regular and Bold available.
However, if you do the same and look at Gulim (a popular font in Korea) you will see that only Regular is available. There is no Bold alternative for bold text formatting to access.