Maybe.
Branch Circuits Required
National Electrical Code 2014
Article 210 - Branch Circuits
I. General Provisions
210.11 Branch Circuits Required.
(C) Dwelling Units.
(1) Small-Appliance Branch Circuits. In addition to the number of branch circuits required by other parts of this section, two or more 20-ampere small-appliance branch circuits shall be provided for all receptacle outlets specified by 210.52(B).
III. Required Outlets
210.52 Dwelling Unit Receptacle Outlets.
(B) Small Appliances.
(1) Receptacle Outlets Served. In the kitchen, pantry, breakfast room, dining room, or similar area of a dwelling unit, the two or more 20-ampere small-appliance branch circuits required by 210.11(C)(1) shall serve all wall and floor receptacle outlets covered by 210.52(A), all countertop outlets covered by 210.52(C), and receptacle outlets for refrigeration equipment.
Exception No. 2: The receptacle outlet for refrigeration equipment shall be permitted to be supplied from an individual branch circuit rated 15 amperes or greater.
210.52(B)(1) Exception No. 2 specifically says that you can put the refrigerator on a dedicated circuit (that can be 15 amperes instead of 20), but does not require it. The rest of 210.52 says that you have to have at least two 20 ampere branch circuits to supply all the required receptacles in the kitchen, and that those circuits can only serve receptacles in that same kitchen (with some minor exceptions).
If you can meet the provisions in 210.52, and still end up under the load calculations in other parts of the code. AND the dishwasher is cord-and-plug connected, then it is possible to have the fridge and dishwasher on the same 20 ampere small appliance branch circuit. Whether or not this is recommended is subjective, and can be debated in another place.
Load Calculations
National Electrical Code 2014
Article 210 - Branch Circuits
II. Branch-Circuit Ratings
210.23 Permissible Loads.
(A) 15- and 20-Ampere Branch Circuits.
(1) Cord-and-Plug-Connected Equipment Not Fastened in Place. The rating of any one cord-and-plug-connected utilization equipment not fastened in place shall not exceed 80 percent of the branch-circuit ampere rating.
(2) Utilization Equipment Fastened in Place. The total rating of utilization equipment fastened in place, other than luminaires, shall not exceed 50 percent of the branch circuit ampere rating where lighting units, cord-and-plug connected utilization equipment not fastened in place, or both, are also supplied.
This means that the refrigerator must be rated less than 80% of the branch circuit rating, for it to be connected to a 20 ampere small appliance branch circuit. This would mean the fridge must be rated less than 16 amperes.
20 amperes * 0.80 = 16 amperes
It also means that the dishwasher (assuming it's not a portable unit), must be rated less than 10 amperes.
20 amperes * 0.50 = 10 amperes
If it's a portable dishwasher, it can be rated up to 16 amperes just like the refrigerator.
tl;dr
So if you have:
- A cord-and-plug attached dishwasher fastened in place that is rated 10 amperes or less.
- Or a cord-and-plug attached dishwasher not fastened in place that is rated 16 amperes or less.
- And a cord-and-plug attached refrigerator not fastened in place that is rated 16 amperes or less.
- And you meet the provisions in 210.52(B)
Then it is possible (though maybe not advisable) to have a dishwasher, and a refrigerator on the same small appliance branch circuit.
No!
The only safe way to increase the circuit's capacity is by replacing the wire with one of adequate gauge. For 20 amps, 12 AWG copper is adequate for up to about 100 feet.
If you simply replace the breaker, the wire can overheat and ignite the building from inside the walls.
To resolve the dimming issue, check that the outlet is in good condition and that the wires are securely fastened and not showing any signs of overheating: blackening or loss of the copper shininess. Also, check the end of the wire inside the service panel, both neutral and "hot" (black) wire. If those are okay, set the appliances to a lower wattage setting or replace them with lower wattage models.
Best Answer
In a kitchen there are a minimum of 2 20 amp circuits required. Where I live a dedicated circuit is required for dishwashers, Most of the houses I have wired have 4 branch circuits or 5. I usually put the fridge on its own circuit, a microwave and the disposal on Thier own circuit, if there is a trash compactor the disposal and compactor. In some cases where a 120 outlet is needed for a gas stove or cooktop I will put that on the same as the microwave. There is not a code requirement for this many circuit but the permits only cost a couple dollars per breaker and a few extra feet of wire. I have had to upgrade quite a few kitchens over the years and only one I installed as the owner was pinching pennies and it cost him almost 3k to do what my proposal was at a 400$ savings up front.