It also depends on the "Derivation Path" that a wallet uses... not all wallets use the same Derivation path, so if you put the same 12 word phrase into two different wallets, you MIGHT get different addresses (and therefore, not be able to get access to your coins)
For instance:
Bitcoin Core: m/0'/0'
Blockchain.info: m/44'/0'/0'
MultiBit HD, Breadwallet, Simple Bitcoin Wallet: m/0'/0
Electrum: doesn't matter, uses a completely different 12 word phrase system
It isn't necessarily the end of the world, as there are tools (https://iancoleman.github.io/bip39/) that will take a 12 word phrase and let you specify any derivation path you like and be able to retrieve the keys... it just may not be as simple as installing "SomeWallet" and entering your 12 words