There's one capacitor that's typically faulty on these. I bought a T1950CT the other day that was doing similar to yours, it would attempt to power up then shut off, and apparently after a few hours it would start working.
Here's my experience with the repair: Re: What retro activity did you get up to today?
I made a capacitor map and this should be common to all of the Toshiba 486 and early pentium laptops like the T19xx, T4400 > T4900. Essentially, the beige Toshibas.
You need to clean off any electrolyte on the board, usually a brown-ish gunk that might have a fishy smell. Use IPA and a brush to clean it off, some can get stuck under other capacitors, but it's pretty much just C511 that fails. You can replace it with one of the same value, though Internet has said that the voltage is a bit low for the job there so if you can fit a higher voltage cap with the same capacitance, it might be good to. I just replaced mine with the same value and the laptop went from unable to boot, back to working reliably.
For taking the laptop apart, do refer to the Replacement Procedures section of the maintenance manual for the T1910 or another one like the T1950. There are lots of screws with different lengths and putting a long screw into a short screw's hole can damage things quite badly. I kept a record of screw lengths in notepad as I went to keep track of everything.
These older Toshiba laptops are quite resilient, but be careful with the ribbon connectors as they were fragile to start with and age has not helped. The ones on mine were all okay though, nothing crumbling to dust. When you're in there, you should also remove the green standby battery in there. These ones don't suffer battery damage so badly as the Pentium-type Toshiba laptops, but they don't work anymore and they will eventually start corroding the mainboard.
Edit for 2024:
Updated the image with more information regarding cap sizes. See this post for more details: Re: Old Toshiba T1900 series laptop faulty PSU boards
Original version of capacitor map:
