Your best bet is to contact the power company and ask for an experienced serviceman/person to make a visit to your home and check. Needs to be an experienced person. Any voltage draw upline or downline has nothing whatsoever to do with your bill. You are billed on your usage not someone else's. Lights getting dim can stem from several different issues, but most likely a loose connection on a hot wire "not" a neutral wire. Lights that get very bright and then suddenly dim stem from most likely a loose neutral connection. Contrary to what some people believe the best way to check this is by loading/burdening one hot wire at a time in other words one of the 120 volt legs "not" the 240 combined. You can burden one 120 volt leg at a time by connecting something with a heavy amperage load to one leg at a time and checking the voltage with a good meter. You must check while the leg is under burden. The power company has such equipment that they can use to check on the source side of the meter. The best device for the homeowner is a simple handheld blow dryer such as your wife might use. Look on the side of the blow dryer it will most likely say anything from 1200 to 1800 watts, that is like having eighteen 100 watt bulbs on all at any one time, and that would be on one 120 volt circuit. That will burden the one hot leg so you can take a voltage reading under load. That would need to be checked on both hot legs, which you can not do if you don't know/understand how your house wiring works. Low voltage coming into the home will not cause a high power bill because it has nothing to do with power usage. Too low of source side voltage can cause equipment damage so it is an important factor to consider. Often times low voltage can be noticed when lights dim simultaneously as the heat pump or central ac comes on. It is not uncommon to see some voltage drop but too much can be a problem. Keep in mind that more often than not the average homeowner is using power when they don't realize it. For example your electric hot water heater is continuously working even when your not using hot water. Your freezer, and fridge is also. Take an amp meter and clip it around the different circuit wires in your breaker box and one can easily tell which circuits are using power even when one doesn't realize it. The power company will most likely not check anything on the homeowner's side of the meter due to liability reasons, but at least you can eliminate any problems on the source side by having them come out and check. Also, for a minimal fee they will usually do an independent meter test so any problems with the meter can be checked. Meters can be defective but it is rare and I mean VERY rare. One can have an additional meter installed in the line going to the breaker box but it will require a qualified electrician and a city/county inspection. The meter will belong to you the homeowner not the power company, therefore you can compare the two meters side by side. It's very unusual but it is done in extremely rare instances.
"Best bet" call and ask for an experienced service person. Sorry for such a long post but believe me when trying to explain this, it's actually a short one.