The fishing line didn't work for you? That worked for me, to an extent, although additionally I got a new box (which by itself didn't work) and also at the same time as putting the fishing line on I moved it's position by just a few feet and aimed the opening just a little bit different.
Last year we had a successful nesting in the old box, but this year I moved it about 15 feet it turns out (I thought it was less) closer to the house and ended up with a sparrow chasing off the bluebirds. (We saw the bluebirds at one point but the sparrow had taken it over.) I started researching and found that I might want to get a different box. I was debating slot opening vs Peterson style box with oval and I finally went with latter after reading this:
That Remarkable Peterson Entrance When I first put it up in the same location, the damn sparrow came back and claimed "his" box again. Argh. So, I went with the fishing line, moved the pole a few feet in the direction away from the house, and turned the opening of the box a little bit (instead of directly north towards the house it now faces about NNE or maybe as much as NE). After a week or two, bluebirds came back and I just checked today and they have a lovely cup nest in there full of pine needles (there's a handy white pine nearby).
This page was helpful in a number of ways although you have to overlook the pictures of dead birds:
Managing House Sparrows That's the worst part about house sparrows, that they'll actually attack bluebird nests. The male house sparrow will claim and be very attached to a box. Once he claims it, you can't do much about it except take it down, move it or do something else that deters him from staying there. The fishing line I did something like the grayscale diagram on that page, as well as the X on the roof, although I never got around to those dangling weighted lines on the edge. The bluebirds have no problem perching up there; I've seen both up there at times, sometimes together.
Being diligent about removing the sparrow nests will only prevent them nesting. You'll never get a bluebird to use that box this year if all you're doing is removing sparrow nests. If you can move the boxes just a little and/or change the orientation of their openings a little, in addition to the fishing line, maybe that would help. If you move them, the direction you want is away from buildings and other such spaces that the house sparrows prefer. It may not need to be much; I backed my pole away no more than 5 feet. With those changes, never saw a sparrow on it since, even though I know the one had already claimed it before.
I don't know where you're located but you may have different results than I if you're looking for western or mountain bluebirds rather than eastern. There are a couple different things to consider as far as the size of the opening of the box, etc.
Since you talk of multiple boxes I assume you are aware of the large amount of space that should be between them. My yard is small enough that I can have one box and that's it.
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