In the past people found that resetting the device does not help, and when downloading the podcasts in iTunes, they then sync to the iOS device without error. This suggests one problem might be with the way the iOS devices are managing their connections with the podcast sources, and MacFixIt reader "David" recently wrote us describing the use of threat management software or transparent proxies on the network as potential causes for this issue.
As David described on his "Nerds on site" blog, some networks use a Unified Threat Management service to actively scan for virus activity and other malware as you use the Internet. If you use a UTM service, then it may be disrupting communication with the iOS device, so you might try adding the podcast URL source to the UTM's exceptions list so the rules will not apply to the podcast's source.
One exception you should add is the "itunes.apple.com" url, but check with the podcast producers for the URL if their podcasts are hosted elsewhere.
The second possible source of problems could be the use of proxy servers by your Internet provider, which are used to distribute network traffic and prevent bottlenecks. I have not seen any instances of this causing connection problems with iPods and iPads, but David mentions a few and has a method of checking whether or not an ISP is using a proxy server. Click this checker link, and if it shows the same IP address for both the HTTP and HTTPS protocols, then your ISP is not using a proxy. But if the IP addresses are different, then your ISP is using a proxy.
While you can request that ISPs do not proxy your connection, doing this might be a bit of an inconvenience and is not guaranteed to fix the problem and may also slow down other network activity.
Currently, the workaround to this problem that has been proven to work is to use iTunes to download and manage podcasts, and then sync them to the iPod.
Overall the iOS's handling of these and other unique network set-ups might be the cause, or at least a contribution to the cause for these odd problems. Ultimately, Apple will have to address the compatibility with these devices in an iOS update, so we will have to wait and see if the problem persists in the next release or update to iOS 4.