To a certain extent yes.
If you're mowing tall fescue to an inch or less that helps to create good conditions for weed seed germination. The aim is to keep it high enough so that sunlight doesn't get down there. Figure out what type of grass you have and google "(grass type) recommended mowing height" to which you will get many results for agricultural extensions publications and you can average them all together and come up with your number.

Try to cut frequently enough so that you're only cutting off about a third of the grass at a time.

Often people say "mow at the highest setting on your mower"... this is really not a good method to decide cutting height IMO unless the person saying this knows what that settting height is.

My guess is that you have several types of cool season grasses in your lawn including Kentucky Blue Grass. KBG will crowd out weeds if watered deeply, given enough nitrogen, and mowed properly. I mow my KBG to 3" when it reaches about 4 1/2".

You said that your next door neighbor has a great lawn. He's a good source of information because he knows your local conditions. You may want to follow his advice until you have a lawn that looks as good as his.

-Deerslayer

Yep. And we're one of those groups that, if you attend the meeting, but are the last to arrive, you'll have to chair the meeting. Last guy in the door has to chair the meeting. And if you don't show up at all, we're likely to make you President.

So you better show up here, read the posts, ask a few questions, regularly, just in self defense, if you know what I mean.

You should plan on putting a 'light' not a 'full' application of fertilizer down, this spring. Sooner the better, as hot weather is on the way. My guess is that your neighbor will be quite happy to provide some advice. You should plan on putting down some Weed B Gone, which will kill the weeds slowly, and provide room for your newly-fertilized grass to start spreading out (which it won't, if it's not fertilized).

Your major application of fertilizer won't be until September or October (and I gotta tell you: your lawn is going to look Amazing next year) but you'll see a wonderful improvement this year. Your neighbor will help, and we'll help too.

Yep, it's true. If dear neighbor has a great lawn, did you think he was trying to sabatoge you? Weeds want the same basics that grass wants, air and water. The only thing is, they want them in different way and utilize them differently. Keep the grass higher than the weeds to shade the weeds from sunlight and smother oxygen. Deep water to irrigate grass roots because weeds are basically shallow-rooted and like shallow water conditions. But, there is more to mowing that best serves the grass plants, themselves.

And don't be afraid of the weedkiller. It is likely you have bluegrass (please correct me if I'm wrong) in your lawn and maybe other species along with it, but the bluegrass, given the opportunity, will fill in the empty spots. One good thing for you is it isn't a good idea to broadcast herbicide anyway (although I must admit I would probably do that being the way you describe your lawn LOL), so just get a trigger spray and spot spray some at a time. Give the grass a couple weeks and spot spray some more. I am suggesting this along with a good maintenance schedule to follow. Scroll down that link for schedule outlined in red. Pay particular attention to the paragraphs on watering and mowing. I feel like a broken record because I say that so often, but it is necessary and very important.

Great advice from your neighbor....and he has the proof of the pudding.
But, many purists do put a bad light on weed killers...
broadleafed herbicides...they can take care of such weeds and wont hurt natural grasses. So what you have, if done in....it doesn't deserve to be there. Let the weed killer take care of it. The grass as it grows in, and allowed to grow taller, will then shade out other weeds, new weeds the herbicide didn't take care of.

But, when spraying any chemical...do go by the rules and do read the label.
Spray on a windless day and if you are of child-bearing age, always....always....wear a mask.

Water the area first...this sets up the weed killer's ability to be drawn down into the soil and not just evaporate on the surface.

Keep pets off the surface sprayed for at least 48 hours or longer.