"I can move it, so it's clearly not broken..." Maybe, maybe not! In primary care settings, the majority of fractures that we treat are diagnosed days after they occur (sometimes weeks), ignored because of this inaccurate belief. Wrist fractures are notorious for this presentation, because often there is not dramatic bruising or deformity, and swelling may not be impressive. Finger, foot, and ankle fractures tend to have much more impressive dark bruising and swelling, but even with these injuries, the affected area can most frequently still be moved voluntarily (albeit with significant discomfort).
Fractures come in many shapes and sizes. Fortunately, most fractures are not "open", meaning most have no break in the skin (no bones sticking out anywhere). Some fractures are simple a hairline crack or a small buckle in the bone, while others break all the way through the bone and may shift out of place under the skin. The only way to be certain whether or not there is a fracture is with imaging- most commonly an ordinary X-ray.
What should you do if you think you may have broken a bone? Is it an emergency? If it is "open" (bone sticking out), of course this is an emergency, and this most likely would have occurred in a very traumatic situation such as a car wreck or other high speed injury. Call 911. If the limb is obviously distorted, or there is new and persistent numbness, tingling or decreased circulation (area turning cold, pale and/or blue), you need immediate medical help. In less traumatic situations- you smash your toe against a door walking to the bathroom at night, you step off a curb funny, you fall on an outstretched hand- your injury may be urgent, but not emergent. Follow the RICE pneumonic- Rest, Ice, Compression (ace wrap) and Elevation while you are waiting for medical care.
BOTTOM LINE: Voluntary movement of a finger, toe, hand, foot or leg does NOT mean a bone is definitely not broken- head to your doctor if you are having persistent pain, swelling or bruising, and find out for sure.