Hi Andrew Buxton,
Indeed, it is interesting that the same code under the same browser behaves differently on different machines. What I can suggest is to try the following:
1) Set explicit width and height to the tooltip, especially if you are using RadToolTipManager with LOD. Note, that if the tooltip should be bigger than you set it is not a problem and it will resize if you have not changed the default value of the ContentScrolling property.
2) Try to call _show (note that this is a private method _show and not the public show) after the tooltip was shown. If this does not help, try to call it with a little timeout (try with 0 and e.g 10); If you are using LOD try to call it in OnClientResponseEnd event.
3) Try the suggested in 2) with the private _adjustCallout() method (again when shown and when loaded events and with/without timeout).
If one of the above resolves the issue but there are still some side effects (e.g jump, etc) let us know what are the results of your tests. If none of the above helps, please let us know how to observe the issue locally because otherwise we are not able to help without knowing what is actually going on.
All the best,
the Telerik team