OK, so I have begun to collect old laptops (only T-series from Toshiba), and I already have T1000SE (80C86 processor), but I lack Linux skills to make ELKS work on it.
Now, I'm going to buy T1600 (286-processor), and I'm wondering is there any Linux that is able to run smoothly on it "out-of-box".
It has plenty HD space, 20Mb.

I don't think that there's going to be an 'out-of-the-box' 286 based distro... the main kernel is 386+ oriented. There are, however, some projects aimed at making it work on the 286... alas the 8086 (186) has been avoided (to the best of my knowledge). I would have thought that 20Mb is sufficient to get a nice basic setup (hell, 1.44Mb seems plenty!), but you're highly unlikely to get anything more than a working Linux with a text-editor.

I can't remember the url for the project(s), but a quick google/linux should reveal all.

Good luck.

Then I just need to stick with MS-DOS 4.01, built in ROM.