The battery is unlikely to be the problem. All laptops I know work fine without the battery. Some laptops may have trouble running with a defective battery - the battery may claim to be ok but fail to deliver enough power - but even that is unlikely.

Most probably, either the power supply is broken, or the laptop itself (probably something on the main board, or maybe even the power button ;-)).

You can try to check the power supply, by either switching in a known good one (if you can get one, e.g. from a friend or colleague), or by measuring its voltage (after looking up the expected voltage). If that is not it, the laptop is defective and has to be repaired.

The difficulty of repairing a laptop ranges from easy to almost impossible, this depends very much on damage and model. At any rate, unless you have experience with electronics repairs, it's unlikely you'll be able to do anything. Try finding a small shop that will give you a quote for a repair at a low price - or just sell the laptop as defective on ebay or similar, you may still get a few dozen $ or € for it.

Remove the battery. Unplug the computer. Press and hold the power button for 30 seconds. Put it back together and try again.

Every once in a while, a laptop seems to have a problem like this. My current thinkpad (W510) does it occasionally, but I learned this trick from HP and have used it on Dell, Compaq, Lenovo, ASUS, and a few others brands I can't recall.

The exact symptoms often differ, even on the same computer. The common symptom is the BIOS doesn't come up at all. Sometimes the power button appears to do nothing. Sometimes it briefly powers parts of the computer. Some times the computer stays on and does nothing.

Theoretically, it completely discharges the motherboard capacitors. Its possible there's another issue, but this is a good procedure to start with if a Laptop doesn't POST.

Discharging didn't work

When discharging doesn't work, then something is phsyically wrong. After doing some research, I see other T60 uesrs have encountered similar issues-- ranging from a bad A/C adapter (and/or bad battery), a bad keyboard, a short in one of the USB ports (inspect it), or another component.

Obviously, if you have some sort of warranty, working with the manufacturer at this point would be a good option. Otherwise, if you want to try more options to fix it, you'd have to find out which component. Check to see if it does anything with just the A/C adapter-- try another one if you can. Same with the battery (with and without A/C adapter).

Inspect the external ports for debris (you may want more light). Once you've exhausted those option, work with a computer geek (yourself or someone else), to disconnect the user-servicable components. Hard Drive, RAM, Wireless Cards-- most of these can be accessed from the bottom. The keyboard and touchpad should added as well. I'd test after each component. (The interesting thing about the RAM is if the RAM is bad, it may not try to POST, but if there's no RAM, it usually gives you a clear POST error. At this point, getting any sort of error would give you more information). If the broken component is user-servicable (like the ones listed above), it should be pretty simple to replace or work without it.

Nothing is left, still doesn't work

If you've tried removing all the "user-servicable" parts (not the CPU, motherboard, or a few other parts) and it still doesn't work. The broken component is likely on the motherboard. It could be a port, a blown capacitor, a bad connector for the power adapter and/or battery, bad BIOS, bad chip (Nvidia's mobile GPU recall displayed similar symptoms). Fixing it is usually not an option (you would know how to do it if it was). Replacing it can be done, but its a very demanding process even for the most skilled computer professionals (most simply won't do it because it eats up too much time... testing for the bad component as I described above often takes too much time).

I'd be looking for any parts missed, cleaning out the ports at this point before giving up on it.

Options are very limited at this point. Before buying a replacement motherboard, you should investigate the other options. If warranty options are available, they may be the most cost effective option. Getting a new laptop may be the most cost effective option. If you'd rather stick with the T61, consider buying one from ebay and using it for parts-- used laptops are often comparable to their used/refurbished motherboard prices, and to be honest I'd sooner replace the screen than the motherboard. The 'right' choice depends on your circumstances.

Replacing the motherboard

If have have several hours to spend over several days you truely want to attempt to replace the motherboard and you don't have any experience with actually doing it, you need to understand:

• You may break it more There are several parts of the laptop that will easily break when you're disassembling it. Its highly likely you won't be able to put it back together correctly, that its very possible some part will be misplaced or lost. These things aren't designed to be user-replaced, don't be surprised to find a part you have to break to remove another part. You may also need specialized gel or other resources to properly put it back together.
• Little to no resources There is almost no standard like on desktop computers and very few resources available for doing this task. Because its a Thinkpad, you have more than most people.
• You're giving up all help Any possibilities of a warranty or asking someone else for help will be nearly wiped out and you'll just about be on your own as most companies that do work on laptops will not work on it if they see that you've taken it apart yourself. (AKA: Last resort, only do this after talking with your other geek friends, see if they want to learn too-- if there's a specialized company that does 'laptop repairs' or 'laptop motherboard replacements' they'll likely be more cost effective).
• It may not work After all that work and money, the problem may simply represent itself because the motherboard went bad because of a bad component you reused.

Learn to take it apart incrementally. The exact process varies, but once you have something off, put it back on. Then take it off and put it back on again. Eventually, when you've taken the laptop completely apart. Put it back together. Put it together at least two times before spending any money-- more if possible. If you can, take pictures every step. It'll be best to have something to organize the screws (there are a lot of screws, especially in a Thinkpad).

Good luck.