I had the same issue with my daughter's Studio 1558 after it was 2 years 2 months old. I replaced the coin battery as someone suggested but nothing changed. My symptom was that I'd insert the power adapter cable into the laptop and the charge light would glow for about 1 second and then turn off - probably because the battery was fully charged. i could verify the status of the battery by pressing the button on the battery - all 5 lights glowed indicating a full charge.
Her laptop has a lighted keyboard which clued me into what the problem was. I noticed that both hindges for the monitor were cracked on the rounded part, so I wondered if there was something lose or wrong with the connection to the power button. I slowly moved the monitor up/down while holding in the power button and noticed the keyboard flash on very briefly. I then tried to find that spot where the keyboard lit up. Once I did, the computer booted up.
So in general, there is a lose connection that causes the power button not to be recognized. Once we find the sweet spot, the button works and it powers up and the computer keeps running no matter what position the monitor is in. So it only seems to affect use of the power button and once it's running, things are fine. I'm not sure if I'll take it apart and see if I can fix that lose connection. Perhaps I'll do that once it dies for good.
She said the cracks appeared within the 1st year, and the boot problem just started happening at about the 2 year mark. I'd say that's kind of a weak point with this laptop - poor craftsmanship. I have several Dell laptops for work that I've used for 4 or 5 years (some are Latitude 830's). I've traveled all over the world with one for 5 years - it took a beating, has a cracked case, a hole in one corner from when I dropped it, yet it still runs fine. So I suspect this issue with my daughters Studio 1558 is a design flaw in that computer - not necessarily every Dell.
My company has since switched to Lenovo for some reason. Those computers seem fine and I've traveled with them. Not sure about durability yet. I just hate where they placed the Function key - non-standard compared to everyone else in the world. Extremely frustrating for someone who works with both Lenovo and other brands at the same time. If I only used Lenovo I guess I'd get used to it, but it drives me crazy!!!!!!!
Hope that helps some of you... good luck!