Hello,
There are two types of hardware handshaking. The most common one is RTS/CTS, but there is also DTR/DSR. Check that the teminal program is set up for the requred one.
The UART in the PC will most likely utilize a FIFO buffer, and the buffer may need to empty before transmission ceases. Therefore the handshake signal should be de-activated well before the buffer is full, to allow for the size of this buffer.
When you say that you "raise the CTS pin", to which side of the RS232 buffer do you refer? Devices such as the MAX232 invert the signal. The active state for CTS will be high on the line side, and low on the MCU side.
Regards,
Mac