I am not even quite sure how this data would be organized. But I am looking for a data source for different speed limits in the US correlated to street name.
The best source at a broad level is from the Federal Highway Administration.
There is a also variety of open data on traffic safety. Some of this has embedded speed limit information as well as other information, such as fatalities and accidents. There are also some state speed zone data zones, such as those for Virginia.
There was a question on another Stack Exchange about this as well: "you can derive an approximate max speed limit by looking at the national speed limit for the type of road in the country of interest" derived from OpenStreetMap.
The DOT National Transportation Library has FAQs about the National speed limit and speed limit laws. Speed limits are governed by state and local jurisdictions. Some roadway data from a subset of states is made available through the Turner-Fairbank Highway Research Center (TFHRC) (or pull a data request). Reference librarians are available to help with specific data requests.
(Disclaimer: I am the Evangelist for Data.gov. A contractor with Dept. of Transportation also provided information for this answer.)
Considering that Google and others estimate travel time, it seems likely that this data may have been compiled (unless they are just guessing by road type). I know of no way to get to this directly but it might be worth pursuing/requesting. I could imagine that they might want to protect it, assuming they actually have compiled it.
Another source of data is http://www.wikispeedia.org. They seem rather interested in speed limit signs but they do have an API.
Otherwise, you may be able to get some government data but smaller roads are generally regulated by local authorities.