How many pieces of armor and clothing can you wear?

There's nothing wrong with Kalina's answer, but here's some more information.

You have 9 equipment slots, not counting weapons:

• Hat
• Eyes
• Mouth
• Clothing
• Chest
• Left Arm
• Right Arm
• Left Leg
• Right Leg

Many pieces of equipment will take up more than one of these slots. Equipping an item will remove any other piece of equipment occupying that slot.

You can tell which slots a piece will take by looking at which parts of the body light up when you select the equipment in the Inventory screen.

Examples:

A bandana will take up the mouth slot. This is compatible with most helmets and glasses.

The Assault gas mask will take up all three head spots: hat, eyes, and mouth. This is true of other closed helmets as well, such as a Synth Field Helmet. Equipping a pair of glasses would unequip these items, as they share the eyes slot.

The Grognak outfit will take up the clothing and chest slots, but leave open everything else, allowing you to wear armor on all four limbs. If you were to put on chest armor, it would unequip the Grognak outfit.

An environment suit will take up all the slots. You will have to put everything back on individually after putting on an environment suit, and equipping any other item will unequip the environment suit.

You can get more information on which slots are left open by specific pieces of armor and clothing on Nukapedia, specifically in the Free Armor Slots column of the Outfits table.