Not in CS5. It probably could be done in CS6 with the new Place and Link command under the Edit menu.
There are a few ways to fake it. I've seen discussion of exporting a selection for InCopy and placing that. Seems to be a pretty close workaround that will do what you want, although when I was playing around with it I had to refresh the target .icml frame. It may be possible to use this workaround in your case, I'm not sure.
You might be able to use a cross reference - but then you couldn't do it by frame, as you requested. Once again, you'd have to click the "Update cross-references" button to update the target, it wouldn't happen automatically. You'd update the text in one area, and then update the cross-references to get it to propogate to the other location. This solution is kind of finicky.
Sometimes, I have turned on "Create Links When Placing Text and Spreadsheet Files" in the File Handling section of the Preferences dialog, so I could have multiple frames all update more-or-less automatically from a source text file whenever I reopened the InDesign file. I like this solution for two reasons. First, I always get reminded to update links whenever I open an InDesign file, so I am always reminded to update the "target frame" unlike the cross-reference method. Second, all of my text content in ID is submitted to me by translators in easily linked text formats (like Word or RTF or raw text or et cetera) so I already am accustomed to managing linked text files.
@fumcarlington – there is quite another way of doing this:
Use a separate InDesign file for the text frame (we could call it "Source.indd") and place that InDesign file on two different pages in your "Target.indd". Every time you update "Source.indd" and open "Target.indd", you will get noticed that a placed file has changed. Click "Update" and you are ready to go.
If that workflow is good for you depends. If you often change the size of your text frame according to other layout elements or if it has to be threaded, you might be better off with an InCopy based workflow.