Ways to Get Bitcoin Test Coins

So I've generated a couple of addresses on the test network using the same mnemonic: tb1pfscdjh8dkqy98gp0xkudrkx88zwslsvd32z60d9nhpm343rewsmqf55g7q and tb1q84jq2rq4jdf8nauum7haneseaqmg89zj944x5k. Receiving test coins at tb1q84jq2rq4jdf8nauum7haneseaqmg89zj944x5k is straightforward, but for some reason, tb1pfscdjh8dkqy98gp0xkudrkx88zwslsvd32z60d9nhpm343rewsmqf55g7q won't accept them. Why's that happening? What are some ways I can get my hands on Bitcoin test coins?

You can use testnet faucets, see here[1] (some of them are no longer functional) and I'm guessing you can't use the second one because the latter is a taproot address and faucets have not updated their systems to send to those addresses yet. [1] https://bitcointalk.org/index.php?topic=5237763.0

A single mnemonic phrase isnt limited to just two addresses. BIP32 and BIP39 standards, established long ago, allow a single set of words to generate a vast number of addresses within your wallet. As OmegaStarScream said, your second address is Taproot. See also https://en.bitcoin.it/wiki/List_of_address_prefixes Mainly through faucets. Or you can ask someone to send/borrow you testnet coins. Or you can simply send it yourself from one address to another. Why do you need testnetcoins on multiple addresses, anyway?

Usually the Testnet faucet https://testnet.coinfaucet.eu/ is working (sometimes this site has issues, if so try later) and will send you some Testnet amount to as far as I remember non-Taproot Testnet addresses (legacy and segwit addresses). If you need Testnet coins on a Testnet taproot address, use a wallet that can send Testnet coins to taproot addresses.

Can I mine test coins? I have some mining machines.

A mnemonic is a seed to generate private keys not addresses. The address is later derived from the public keys derived from those private keys by the wallet you are using and can be of any type. You can, but since the block height is too high (2.5 million blocks) the block reward at this point is very small at about 0.012 (TestNet has had far more halvings than MainNet). Just receive the coins from faucet into a different address then send them to your taproot address. Or you could just use RegTest.

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.