Taste is extremely subjective so one persons idea of a good vodka might not be the same as another persons; however, in the United States, vodka is legally defined as,
Neutral spirits distilled or treated after distillation with charcoal
or other materials so as to be without distinctive character, aroma,
taste or color
This gives us a fairly unique bar since it shouldn't taste like much of anything if the product is being sold as vodka. Some manufactures of vodka will filter it (one, two, three) using activated charcoal and industry grade products are sold for this purpose. Activated charcoal is what BRITA filters contain so from that standpoint the claim is plausible and in fact the examiner.com explained how charcoal filtering is used in industry,
How is it done? It’s quite a simple process really. The picture
accompanying this article shows a charcoal filtration cylinder used at
the Great Lakes Distillery in Milwaukee, Wisconsin, to make Rehorst
Vodka. They pack the cylinder with fresh granular charcoal then drip
the vodka through the cylinder.
The result is a cleaner, smoother tasting vodka with a more pleasant
aroma, and not incidentally a drink with fewer traces of heavy
alcohols that might contribute to that queasy feeling the next day.
Many vodka producers claim a certain number of passes through the
charcoal filtering cylinder on their labels. Great Lakes Distillery
doesn’t do that; their philosophy is to filter it as many times as
necessary to make the vodka just the way the master distiller thinks
it should be.
When does the charcoal get refreshed? When the master distiller thinks
it is time to do so; that varies from distiller to distiller.
With regards to taste, MythBusters has conducted a double-blind taste test with three testers to include Anthony Dias Blue as expert and found that subjectively the filtered vodka did taste progressively better; however,
They analyzed the vodka samples and found that there was no difference
in chemical composition between the filtered vodka and the unfiltered
vodka. You're better off buying the top-shelf stuff than wasting a
bunch of water filters.