MIDI and DMX together
Yes you can use multiple UART/Serial devices with a single Arduino. 

However, there are some things to consider:

The Arduino Uno supports a single hardware serial connection (which is used for programming and debugging with the Serial Monitor). The Arduino Mega supports 4 hardware serial connections. 

The arduino environment has a software Serial library that allows for additional Serial connections. However it is worth noting that software serial is slower than its hardware counterpart. 

Also, if you want to use software serial, be sure to verify that any shields you want to use will support using a software serial connection.