You can download the manual here: http://support.brother.com/g/b/manualtop.aspx?c=us&lang=en&prod=hf_se350eus
That should help make sure you're using the correct size bobbins.
Does this have a vertical bobbin?
If so, there are a few things you can try. Cleaning the bobbin case and bobbin area is key, but you've done that. A defect in the needle can cause this, so try a brand new needle. Modern fabrics are hell on needles and I find they have to replaced much more often than my budget cares for, but it saves a lot of problems.
Also, check every thread guide on the machine, everywhere. Sometimes they can be loose, or have a nick, which can trap thread and cause this type of problem. Unthread, check each guide closely, use a magnifying glass. If any are loose, tighten them. If you find a nick, a bit of plumber's Teflon tape over the spot may help. Or try to buy a new guide. You can even remove it and try to file it smooth if it allows that.
Another thing; Bobbin cases can become so tightly tensioned that the thread can't move. This may be fixable with some tinkering, but take care.
Look at the bobbin case - you'll see it has a very small screw on it. This is the tension adjuster. First, using a Sharpie or something like it to use so you can very clearly mark the exact position of the screw as it is now, in case you need to restore it.
Next, dangle the bobbin case by the thread tail and give it a shake, like you were playing with a Yoyo. If the tension is OK, the bobbin case should slide down the thread slowly but smoothly with each shake. If it just flies down the thread, it's way too loose. If it doesn't move at all, it is much too tight.
You can try to adjust the tension by moving that screw. Move it very, very slightly. Test after each adjustment of the screw. Not meaning to be repetitive but only use tiny, tiny movements of the screw each time, there isn't that much play in it.
With luck, you find the sweet spot for tension. If not, you may have to buy a new bobbin case. I always keep a spare on hand to use for extra heavy threads, like button thread. I keep the spare's tension loose so it'll work with those coarse threads.
Hope something helps.