In almost all cases, this problem is due to errors in the feed itself. FeedDemon goes to great lengths to detect and work around errors in feeds, but there are some errors that can only be resolved by the publisher of the feed. You can quickly determine whether this is the case by right-clicking the feed, selecting "Feed Properties," then switching to the Advanced tab and clicking the "Validate" button.
Of course, sometimes the web server hosting the feed is the cause of the problem. For example, the site may be down, or it may be responding so slowly that FeedDemon "times out" when trying to access it. To view a report of server-related problems that FeedDemon encountered when updating your feeds, select View > Reports > Feeds With Errors. If several of your feeds are timing out, you may want to increase how long FeedDemon waits before timing out - to do this, click the Connection button in FeedDemon's options.
Note: If the feed is in a synchronized folder, it should always receive new articles as long as they're being received by Google Reader itself. You can check the feed on Google Reader's web site to see whether Google Reader has stopped receiving new articles.