Can you mix metals in your finishes? The answer is a resounding YES!! You can also match all your finishes. I personally find that a space can look somewhat boring if you choose to slavishly match every metal finish, but that's me. I like harmony with some variety.
Here are a few guidelines that will ensure that you have a cohesive look.
Work with what you like best
If you are renovating/ updating, choose the finish you like the best / is the most stylish and match to it. Be aware that a dull gold/brass look is trending right now as is chrome, and it may not be long before all that brushed nickel will have a new status.
Limit the variety of tones
If you already have things in silver tones (brushed nickel, pewter or chrome) consider mixing them with wrought iron or black finishes. Two to three tones would be the limit or you will end up with a hodge podge.
Connect with other tones in the room
In the picture below black is used predominately in the light fixtures while the faucet has silver tones and the handles are either pewter or brushed nickel, still in the silver family. Because black carries such visual weight it works best when there is something else black in the space to help anchor it.
Whitney Blair Custom Homes
Many older homes have brass or copper fixtures that look good when paired with oil rubbed bronze or black. If you already have something that is dark in the room this is the tone you should select to supplement with. While copper and black breaks the keeping to the same tone guideline, it can work when you have other black furnishings.
Elaine Griffin Interiors
Copper in this space is paired with stainless steel and possibly pewter. The copper is picking up the tones in the wood and your eye is moved around the space very nicely. It's all a matter of creating flow.