Here is something you could do but you should do it only if you are completely comfortable and sure that you can do it safely.
If you choose to do this, you are completely on your own.
You should remove all jewelry (such as rings, watch, bracelet, necklace) and get a good pair of heavy rubber or clean and dry leather gloves. Wear some kind of eye protection. There may be others on the forum who will suggest some specific personal protection equipment or procedures.
The measurements are to discover any voltage drops across connections that would indicate bad connections or circuit breaker conditions. You are looking for voltages on the order of tenths of a volt where they should be very close to zero, or differences of tenths of a volt where they should be the same.
1. Remove the cover from your service panel.
2. Turn on significant loads so that you get a lot of current. Since the largest loads are probably 240 Volts you can turn those on for one set of tests on both hot sides and then try to set up an unblanaced load (see step 5) for the neutral test.
3. Measure the voltage drop between the conductors coming in from the utility and the conductors going to the largest loads that are turned on for that phase. Any significant voltage drop indicates a connection problem somewhere along that line, or a bad breaker.
4. Measure the voltage between the load conductors after they leave the breaker, and the utility neutral. The measurement should be the same on both sides of the bus. Any significant difference could indicate a utility transformer problem or a connection problem.
5. Try to turn on 120 Volt loads on one side to produce a large current in the neutral. Then measure the voltage drop between the neutral conductor for the circuit and the neutral conductor from the utility. Anything greater than a few hundredths of a volt indicates a bad connection.
6. With the high neutral current condition of Step 5, measure the voltage between the utility neutral and the ground wire in the panel. Any difference indicates a bad connection in the panel, since both are supposed to be connected in the panel.
When measuring things that should be zero (voltage drops on the same phase or bus), use a low scale on the voltmeter. You are trying to get accurate measurements on the order of hundredths or tenths of a volt.