The ST USART will halt DMA if a receive error occurs. Do you have interrupts enabled for USART RX errors, and do you check/clear the USART status if an error does occur? If not, you will never get past the first time a parity/framing/overrun error occurs.
I've seen this many times, where clock jitter in the baud rate clock causes a framing or overrun error. Putty on a PC isn't using the same baud rate generator as the ST, so the fact it works with a PC would tend to point to the USART clock rate. Are you using HSI for the clock source? The oscillator tolerance is significant at high USART baud rates.
