Any character with Alchemy skill can combine potions/powders, it's just a matter of having high enough skill for the recipe you're attempting.
As you guessed, if the game tells you that the items can't be merged, it means that a combination of those items isn't possible. If you're getting the error message "XXX cannot merge these items" and you are using a character that has, or can train the Alchemy skill, you need to raise your skill before you can combine those items.
You can experiment to figure out the recipes yourself, or if you want a list of recipes other people have found, check out the following URL (just see the Gadgeteer/Alchemist Recipe spoiler link). It seems to be a site that compiles information from various forums and has some useful info.
http://www.geocities.com/jandrall