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.
{
    int confirms = wtx.GetDepthInMainChain();
    entry.push_back(Pair("confirmations", confirms));
    if (wtx.IsCoinBase())
        entry.push_back(Pair("generated", true));
    if (confirms > 0)
    {
        entry.push_back(Pair("blockhash", wtx.hashBlock.GetHex()));
        entry.push_back(Pair("blockindex", wtx.nIndex));
        entry.push_back(Pair("blocktime", mapBlockIndex[wtx.hashBlock]->GetBlockTime()));
    } else {
        entry.push_back(Pair("trusted", wtx.IsTrusted()));
    }
    uint256 hash = wtx.GetHash();
    entry.push_back(Pair("txid", hash.GetHex()));
    UniValue conflicts(UniValue::VARR);
    for (const uint256& conflict : wtx.GetConflicts())
        conflicts.push_back(conflict.GetHex());
    entry.push_back(Pair("walletconflicts", conflicts));
    entry.push_back(Pair("time", wtx.GetTxTime()));
    entry.push_back(Pair("timereceived", (int64_t)wtx.nTimeReceived));
    
    std::string rbfStatus = "no";
    if (confirms <= 0) {
        LOCK(mempool.cs);
        RBFTransactionState rbfState = IsRBFOptIn(*wtx.tx, mempool);
        if (rbfState == RBF_TRANSACTIONSTATE_UNKNOWN)
            rbfStatus = "unknown";