Downsides of higher order filtfilt's
The sharper the filter is in the frequency domain, the longer the impulse response will be. This typically leads to "time blur" or "ringing" in the time domain. In addition, a zero phase filter is non-causal, so you get "pre-ringing" and any sharp onsets or transients in the time domain get degraded.

The long impulse response also leads to a long "transition" region at the beginning and the end of the signal. As a very rough rule: if you have a filter that does something drastic over 1Hz of bandwidth, it will need about 1 second of time domain signal until the effect of the filter is fully established. 

The optimal choice is very dependent on the specific requirements of your application.