Does the light switch feel any different than the fan switch? The switch can feel funny if it fails.
Also, what kind of fixture it is? It's probably not relevant, but it's not a bad detail to add.
OK, looking at your diagram, I'm happy to see that all the gnd and all the neutral are tied together. That really helps, as it means you don't have anything too weird going on.
I'm going to describe what I see in that diagram, so that we're on the same page (or so that you know I have no clue).
From what you've drawn, I assume that the wire going to the light fixture is the one on the left, the fan is the one on the right.
One of the center 3 lines goes back to the breaker. The other two of the center three lines are actually branches that feed something else (basically, they are chained off of here). That big bundle of 4 wires on the one leg of the light switch are just a way to get everything tied together without a separate wire nut.
At this point, as long as all connections to the switch are good, and you think the fixture is good, then I'd say it would have to be the switch itself - they are mechanical, and they do go bad over time. If you have a multi-meter, or some kind of circuit tester, then you should be able to check the power at screws of the fan switch - the one coming from the light switch will be hot all the time, the other side will only be hot when turned on. Then try the light switch - I bet it's not actually working correctly.
If you don't have a tester, just replace the light switch. They don't cost much, and then you'll know for sure.
Assuming the light itself isn't bad, then the combination of a dim light and the dishwasher making it go out sounds like a break in a wire or a corroded connection. A break or corrosion can allow enough current to pass to light the bulb dimly, but when a higher current is called for the resistance of the break or corrosion goes up, causing an apparent total failure.
See https://diy.stackexchange.com/a/27285/82
Best Answer
Just a hunch, but worth investigating.
I suspect you have this light fixture somehow linked into two circuits. That should never happen, but if it does, you can end up with a lot of strange things.
Find the Breakers
You need to figure out which breaker controls the switch and which breaker controls the light separate from the switch. Doing that may get a bit tricky, as I would not connect the switch back to the light again until this is straightened out.
I would get a non-contact tester and/or a multimeter to test the switch wires. You should see one of the wires hot (lit up with the non-contact tester) and 120V (assuming US for the moment) between the wires. When you turn off the relevant breaker you should no longer see a hot wire and the voltage between the wires will either be zero or some relatively low phantom voltage.
When you have the switch breaker off, does the light go out? If it does, then the next step is to check each other device (light, switch, receptacle, built-in appliance) on the same circuit (i.e., everything that goes off when you turn off that breaker), looking for loose wires, things wired incorrectly, etc.
If the light stays on when the switch breaker is off then you have to keep flipping breakers until you find the one that turns off the light. Then you know that you have a cross-connection between the two circuits which is NOT a good thing. That could take take several forms, and will likely require checking out the wiring on both circuits. But I would actually start in that case with the wires going into the problem light fixture. One possibility is that wires for the 2nd circuit pass through the ceiling junction box and that when the light fixture was replaced, the wires got messed up. But there are other possibilities as well.