If I unlock an encrypted wallet for 359711509437336256 or less seconds, I can sign transactions with it:

    $ bitcoin-cli walletpassphrase "$x" 359711509437336256
    $ echo $?
    0
    $ bitcoin-cli signrawtransaction $tx | grep complete
      "complete": true

But if I unlock it for 359711509437336257 or more seconds, it appears to have unlocked but I can't sign any transactions:

    $ bitcoin-cli walletpassphrase "$x" 359711509437336257
    $ echo $?
    0
    $ bitcoin-cli signrawtransaction $tx | grep complete
    error code: -13
    error message:
    Error: Please enter the wallet passphrase with walletpassphrase first.

I don't know the significance of that number. It's nowhere near a power of 2.

I was trying to unlock the wallet effectively "forever", so put a big number. If I use too big a number (2^63 or more) I get an error message `JSON integer out of range` but for numbers between 359711509437336257 and 2^63-1 inclusive I get no error, but no actual unlocking.
{
    CWallet * const pwallet = GetWalletForJSONRPCRequest(request);
    if (!EnsureWalletIsAvailable(pwallet, request.fHelp)) {
        return NullUniValue;
    }
    if (request.fHelp || request.params.size() != 0) {
        throw std::runtime_error(
            "walletlock\n"
            "\nRemoves the wallet encryption key from memory, locking the wallet.\n"
            "After calling this method, you will need to call walletpassphrase again\n"
            "before being able to call any methods which require the wallet to be unlocked.\n"
            "\nExamples:\n"
            "\nSet the passphrase for 2 minutes to perform a transaction\n"
            + HelpExampleCli("walletpassphrase", "\"my pass phrase\" 120") +
            "\nPerform a send (requires passphrase set)\n"
            + HelpExampleCli("sendtoaddress", "\"1M72Sfpbz1BPpXFHz9m3CdqATR44Jvaydd\" 1.0") +
            "\nClear the passphrase since we are done before 2 minutes is up\n"
            + HelpExampleCli("walletlock", "") +
            "\nAs json rpc call\n"
            + HelpExampleRpc("walletlock", "")
        );
    }
    LOCK2(cs_main, pwallet->cs_wallet);
    if (request.fHelp)
        return true;
    if (!pwallet->IsCrypted()) {
        throw JSONRPCError(RPC_WALLET_WRONG_ENC_STATE, "Error: running with an unencrypted wallet, but walletlock was called.");