I have a simple armature that has three loose bones that I would like to join at one joint. Is there a way I can join these 3 into a single joint?
You can join multiple bones together. You will need to flip one of the bones around (AltF) so that there is a bone tail in the correct spot. Then just set the other two bones' parent to the flipped bone; and enable Connected.
Create a control bone; Make it small. Basically you need to create an extra bone that all of the others connect to. Although you must create it, you never have to use it in posing. (It can still be useful.) The small end of this bone needs to be facing toward the other bones.
Here is a picture of the control bone created properly and positioned with the little end at the spot that you want the new joint to be.
Create the master joint; Turn on Vertex Snapping down at the bottom of the screen; this will allow you to position the bones perfectly at the location of the master joint. Here’s a picture to help you find vertex snapping.
By default the snap mode is increment. Change this to vertex by clicking the option tab and selecting the vertex option (cube with white dots).
Select one of the joints.
Mouse over the small end of the control bone, and press G. Move the big joint to the little end of the control bone.
Repeat this for the other two bones.
Parent the bones; If you've followed the steps alright, you should have something that looks like this:
Select the three bones that you want to make one joint of. Then select the control bone. Press Ctrl+P>Connected
Boom!; You just made one joint out of three!
It appears that this is not possible unless they are part of the same chain. As the Blender Wiki states:
You can merge together several selected bones, as long as they form a chain. Each sub-chain formed by the selected bones will give one bone, whose root will be the root of the root bone, and whose tip will be the tip of the tip bone.
For further information, read the original documentation on the Blender Wiki.
The snap functions can be used in Edit Mode to set head or tail of the bones. Shift-S Cursor to selected and Selection to cursor could be used to align them.
You can join bones like any other object using Ctrl-J after that the Parent-Control in the bones properties will be available.
Everyone here assumes that the original poster doesn't have the bones aligned the way he wants them. Maybe he does--it is correct in some cases: for example, legs go down and the spine goes up--therefore, the heads of the leg bones aren't positioned over the tail of the parent (but either over the head of the root bone or off to the sides). Also, everyone other answer moves the bones when setting the parent, but that is not necessary. If desired, move the joints over each other (such as by selecting all three [joints not entire bones] then pressing S, 0, enter). Then, to keep them in the same place, shift click (right-click in <=2.79) to select each child, then the parent last, release shift, Ctrl+P (shortcut for Armature, Parent, Make), and click "Keep Offset" (which will not mess up the position like "Connected" or like choosing a parent from the bone tab would). In this case, no, you cannot make them use a single joint, but pose mode will behave as though they do.