You can, check these guides
[Guide] Solo mine testnet bitcoins with bfgminer, Bitcoin Core, and a CPU/GPU (i personally tried this guide with good result).
[Guide] Solo mine testnet bitcoins with cgminer, Bitcoin Core, and a Compac F

You also can use mining pool for testnet coin if you don't want to run your own node, but i don't know which pool for testnet coin still alive today.