LAN speed has no bearing to the internet speed (unless you have +100Mbps internet, which you do not). If speedtest gives you 400 but your downloads are slow, I would wager a guess there's something up at the other end - whether it be slow, overloaded, or just bad servers. Some servers/download sites may also limit your download speed (rapidfail and failupload pop in my mind first, though I'm not sure how they're currently doing, I've had nothing to do with either for years).
Are you downloading from a website or are you using some kinda P2P/Torrents?