Maybe. This all assumes that the hard drive is ok and it really is a problem with other hardware.
It all depends on what types of hardware you have. If it is an older hard drive it will be an IDE or PATA connection and if it is a newer type it will be the SATA connection.
The older IDE/PATA style used a big 4 pin power connector (usually white) and a wide (about 2 in.) flat 40 or 80 conductor ribbon cable with 40 pin connectors.
The newer SATA style has a different style 4 or 5 wire power connector (usually black) and a thin (about 1/2 in) serial cable that only has 7 pins.
For either style hard drive there may be about 4 screws securing the hard drive in the case. For temporary copying, you don't need to mount the old drive in the new case. Just be sure that none of the electronics touch anything and the hard drive doesn't touch anything. I often let them hang on the wires or use a thin booklet or tablet under them depending on case configuration.
If you have the newer SATA connection on the old drive, then no problem. There should be an unused connector on the motherboard and an extra power plug. Worst case scenario here might be to unplug the DVD drive and temporarily plug in your old hard drive. If you really want to boot from it, then you could unplug the hard drive and plug in your old one.(see below)
If the old drive is the older IDE/PATA then the new computer may not have a PATA connector and the power supply may not have the older style 4 pin power connector. If you don't have compatible connectors on the new computer, try and find a friend who does have a compatible computer to copy the data. Another option would be one of the kits that turn a hard drive into a portable hard drive using a USB connection though it might have some issues with the existing partition(s) and format of the partition(s).
If you only passworded the account, then there will be no problem copying the data as a slave or secondary drive. In fact, I don't need to boot Windows or even need your password to read the entire contents of your hard drive. This is how easy it is to bypass account passwords if you have access to the machine. The only exception is if you enabled the encryption on any files or folders. If you enabled the encryption then you'll have to boot the drive and access your data through your account.
If you are forced to boot the drive, then there are a number of driver issues that you may have problems with. None of the drivers installed on the old hard drive will be compatible with the new computer. This could force safe mode operation and might cause problems copying stuff. You may need the Windows disk to install even partially working drivers.
If it is actually a hard drive problem, then the drive is done and it probably won't work in any computer. If this is actually the case, a new hard drive and OS install would fix the old computer.