My Lenovo T60 will not boot up

It's time for you to replace your battery.

how old is your laptop? and how do u use it - do u leave it plugged into the wall socket all the time? It's common for batteries to die out after a couple of years. however, if the computer overheats or if you leave it plugged in all the time, it's life cycle will decrease. Charge - discharge cycles. That's how you are suposed to do it.

a battery will not hold full charge when its not being used it will slowly lose charge over time keep your battery in a dark cool place if it is connected to your thinkpad it will gradually lose charge
hope this helps

I had the same problem. This procedure from the "Technical Questions" section of http://shopdelphi.com/consumers/support/faqs/myfi/ worked for me:
"My MyFi won't power up, how do I reset it?
To reset your MyFi, start by turning the receiver off and removing it from its (car, home, audio system) cradle. Then remove the back cover and battery. Using a small paperclip, press the reset button located on the top left corner of the battery compartment. Now plug the AC power adapter into a wall outlet and plug the power port directly into the bottom of the MyFi (battery stays removed). Wait 10 seconds and listen for a beep. After you hear the beep the message "insert battery" will appear. Cycle the MyFi to ensure it powers up and powers down. If the power cycle is normal, then turn off the MyFi, unplug the power adapter, and reinstall the battery. Re-plug the AC power adapter directly into the bottom of the MyFi. Wait 10 seconds and listen for a beep. After the beep you will see the message "battery charging". Leave the MyFi plugged in until it's fully charged. You will see the message "charging complete" displayed. After the MyFi is fully charged, turn it on and reset to the factory defaults. Unplug the MyFi from the AC power adapter and using the battery power leave it on until it fully discharges (approximately 2-5 hours). Plug the MyFi in using the AC power adapter to recharge the unit and resume normal use. If you need help at any point during this process contact us at 1-877-463-3574 for technical support."

I hope it works for you too.

Boot up in safe mode (press F8 0on startup) on your normal login name. Once in safe mode disable the screensaver, then go to power options and disable them so that there are no powersaving features enabled.. Then restart into normal windows, and hopefully everything will be fine.

the battery of your ideapad is already defective, try to change battery

If the docking station charging doesn't work either and your battery is ok you have lost voltage regulator (it prevents overcharging of battery) Take the battery to a battery plus store and they can check it..

the battery might be so dead that it's faulting the charging system, remove the battery, remove the power then plug it back in (with out the battery) and try turning it on,.. if that doesn't work try looking up how to reset the bios (or remove the bios battery) for a BIOS reset.

Remove the 1 gig chip. Pull you battery out and unplug the laptop. Hold the power button in for 10 seconds. Put the battery back in and plug the laptop back in again and try booting it up. This should reset the BIOS and allow it to boot. The beep codes you are talking about are a memory beep code and it is normally caused by adding memory or a loose memory stick. Also try pulling the existing memory and resetting it.

Try this:

With the AC adapter and battery installed and it will not power up remove the adapter and the battery. Then plug in the
AC adapter after both have been removed for 30 seconds and let the laptop set for 30 seconds and then unplug the AC adapter and then slide in the batter within 15 seconds and try to start the laptop.