static Semaphore requestLimiter = new Semaphore(500, 500);

...

String GetResultFromAPI() {
    if (!requestLimiter.WaitOne(0)) return null;

    try
    {
        // ... code to perform API call, with a timeout specified on the HttpWebRequest
        return result;
    } catch { /* ignore exceptions */ }
    finally
    {
        requestLimiter.Release();
    }
    return null;
}