Twitter API code throws an exception exactly every 12 hours

Exception type: System.AggregateException
Exception message: One or more errors occurred.
Exception stack trace:
   at System.Threading.Tasks.Task.WaitAll(Task[] tasks, Int32 millisecondsTimeout, CancellationToken cancellationToken)
   at System.Threading.Tasks.Task.WaitAll(Task[] tasks, Int32 millisecondsTimeout)
   at System.Threading.Tasks.Task.WaitAll(Task[] tasks)
   at Niknak.Social.Twitter.UserTimeline`1.GetTweets[Tweet,User,Media,MediaSize](String screenName, Int32 count, Boolean includeRTs, String sinceMessageId)
   at NiknakV3.Business.Services.TaskScheduler.Processors.TwitterFeed.ProcessTask(SimpleTaskConfig`1 config, AuditLog currentLog, Boolean& terminate)
Inner exception message: One or more errors occurred.
Inner exception stack trace: 
   at System.Threading.Tasks.Task`1.GetResultCore(Boolean waitCompletionNotification)
   at System.Threading.Tasks.Task`1.get_Result()
   at Niknak.Social.Twitter.UserTimeline`1.<>c__DisplayClass3`4.<GetTweets>b__0(Task`1 requestTask)
   at System.Threading.Tasks.ContinuationTaskFromResultTask`1.InnerInvoke()
   at System.Threading.Tasks.Task.Execute()
Inner exception message: An error occurred while sending the request.
Inner exception stack trace:
Inner exception message: The request was aborted: Could not create SSL/TLS secure channel.
Inner exception stack trace: 
   at System.Net.HttpWebRequest.EndGetResponse(IAsyncResult asyncResult)
   at System.Net.Http.HttpClientHandler.GetResponseCallback(IAsyncResult ar)

public static List<Tweet> GetTweets<Tweet, User, Media, MediaSize>(string screenName, int count, bool includeRTs, string sinceMessageId)
	where Tweet : IMessage, new()
	where User : IUser, new()
	where Media : IMedia, new()
	where MediaSize : IMediaSize, new()
{
	if (string.IsNullOrEmpty(screenName))
		throw new ArgumentNullException("screenName");
	if (count <= 0)
		throw new ArgumentNullException("count");
	if (string.IsNullOrEmpty(sinceMessageId) || sinceMessageId == "0")
		throw new ArgumentNullException("sinceMessageId");

	List<Tweet> outList = new List<Tweet>();
	List<JsonObject> tweets = new List<JsonObject>();

	using (HttpClient client = new HttpClient(new OAuthMessageHandler<OAuthSettings>(new HttpClientHandler())))
	{
		if (client == null)
			throw new Exception("Unable to initiate HttpClient object");

		string requestUrl = string.Format(
			APITIMELINEURL,
			screenName,
			count,
			includeRTs.ToString().ToLower(),
			sinceMessageId
		);

		Task retrieveTweets = client.GetAsync(requestUrl).ContinueWith(requestTask =>
		{
			//get HTTP response from completed task
			HttpResponseMessage response = requestTask.Result;

			//check that response was successful or throw exception
			response.EnsureSuccessStatusCode();

			//necessary when retrieving more than about 15 tweets at a time
			MediaTypeFormatter.SkipStreamLimitChecks = true;

			//read response asynchronously as JsonValue and dump into list for further processing
			Task processTweets = response.Content.ReadAsAsync<JsonArray>().ContinueWith(readTask =>
			{
				JsonArray statuses = readTask.Result;
				foreach (JsonObject status in statuses)
					tweets.Add(status);
			});

			Task.WaitAll(processTweets);
		});

		Task.WaitAll(retrieveTweets);
	}

	if (tweets != null)
		foreach (JsonObject tweet in tweets)
		{
			//we need to do some checks on the message retrieved to ensure it's not fubar
			if (!Utils.IsValidTweetJson(tweet))
				continue;

			outList.Add(Utils.ParseTweetJson<Tweet, User, Media, MediaSize>(tweet));
		}

	return outList;
}