I'm trying to add a 20 amp circuit breaker into my existing main panel. The main panel has one vacant slot to put the breaker, but the neutral and ground bars have no more vacant slots. My question is this -- can I hook up the new neutral and ground wires to an existing slots? Meaning if the slots are already being used by another breaker, can I also hook up my new neutral and ground wires into these slots since there are no more available slots? I also see that the neutral and ground wires are all mixed up on the two bars. I'm saying that neutral wires and ground wires are not hooked up seperately from the two bars. I guess they're okay if done in the main panel.
110.14 Electrical Connections. (A) Terminals. Connection of conductors to terminal parts shall ensure a thoroughly good connection without damaging the conductors and shall be made by means of pressure connectors (including set-screw type), solder lugs, or splices to flexible leads. Connection by means of wire-binding screws or studs and nuts that have upturned lugs or the equivalent shall be permitted for 10 AWG or smaller conductors. Terminals for more than one conductor and terminals used to connect aluminum shall be so identified.
So if more than one conductor can be placed under a lug, it must say so somewhere on the panel (or in the panel documentation).
408.3 Support and Arrangement of Busbars and Conductors. (D) Terminals. In switchboards and panelboards, load terminals for field wiring, including grounded circuit conductor load terminals and connections to the equipment grounding conductor bus for load equipment grounding conductors, shall be so located that it is not necessary to reach across or beyond an uninsulated ungrounded line bus in order to make connections.
If you have a bus bar on each side of the panel like in the diagram above, you should not connect the neutral to one bus and the ground to the other. Neutral and ground from all circuits on the left should go to the left bus bar, and Neutral and ground conductors from the right should go to the right bus bar.
408.41 Grounded Conductor Terminations. Each grounded conductor shall terminate within the panelboard in an individual terminal that is not also used for another conductor.
Exception: Grounded conductors of circuits with parallel conductors shall be permitted to terminate in a single terminal if the terminal is identified for connection of more than one conductor.
So if the panel allows multiple conductors under a lug, you can terminate multiple equipment ground conductors under a single lug. However, you cannot use a single lug for multiple grounded conductors (neutrals), or a mix of equipment ground and grounded (neutral) conductors.
Here's why you can't have multiple neutrals in a single terminal.
Multiple neutral conductors in a single termination create a significant problem when the circuit needs to be isolated. In order to isolate the circuit, the branch breaker is turned off and the neutral is disconnected by removing it from the terminal. If the terminal is shared with another circuit, the connection on the other (still energized) circuit will be loosened as well. Loosening of the second neutral (loss of neutral) under load is a safety hazard, and may establish an overvoltage condition on lighting and appliances if the neutral is part of a 120/240 Vac multi-wire branch circuit. Source
And this is why you can't have a neutral and a ground in a single terminal.
The connection of a neutral and equipment-grounding conductor in the same termination creates a similar issue. One of the objectives of the particular arrangement of bonding jumpers, neutrals and equipment grounds is to allow circuit isolation while keeping the equipment grounding conductor still connected to the grounding electrode (see UL 869A - Reference Standard for Service Equipment). When the neutral is disconnected, the objective is to still have the equipment ground solidly connected to the grounding electrode. If both the neutral and grounded conductor is under the same terminal, this cannot be accomplished. Source
You can , however, have both grounded conductors (neutrals) and equipment ground conductors connected to the same bus bar in the main service panel if the grounded conductor bar/bus (neutral bar) is bonded to the equipment ground bar/bus (it's different if you are dealing with a sub-panel, since the bar/bus will not be bonded).
Long story short
You'll have to check the panel documentation to determine if multiple conductors can terminate under a single lug. If they can. You can connect a couple equipment ground conductors to a common terminal, which should free up enough space to add the breaker.
NOTE: This only applies to bus bar terminals, most breakers are not rated to be "double tapped". So you should never have two conductors under a single breaker lug.
Yes, it's fine that the neutral and ground wires are intermixed. The ground and neutral bar and both bonded inside the service panel, so even if all of the ground wires go to the ground bar and all of the neutral wires go to the neutral bar, they are still electrically connected via the bonding jumper. It is only once they exit the service panel that they must remain separated for various reasons. For appearance reasons, however, it definitely looks tidier if the neutral and ground wires go to their designated bars.
To answer your main question, no, it is not usually acceptable to have two wires go to the same screw on the neutral or ground bus bar. The only exception to this case is if the bus bar is UL-listed for that purpose. Call the manufacturer if you want to find out, or you can just play it safe and use wire nuts to combine the two wires into one wire, and then connect that single wire to the bus bar.