The people picker field in SharePoint 2010 doesn't show any suggestions, even when it says it found matches to your query. I didn't test this thoroughly, but this seems to be the cause at least in the summary links webpart when you add a new link.

I tried clicking it as the error message stated. Even the tooltip mentioned there were available solutions.

I did click it and nothing happened. I became desperate, clicked it like crazy and then from some magical combination of clicking and typing the field height became bigger and this was visible:

Yes... the dropdown opened inside the text field. After inspecting the HTML and CSS I found this solution: 

body div.ms-inputuserfield {
    overflow-x: visible !Important;
}