Well this largely comes down to one's definition of "frame".
I don't think the PM or police ever set out to "frame" Knox, Sollecito or Lumumba in the classical sense of "decide without foundation to pin the blame on....". What I think happened was that the PM and police came to a number of incorrect conclusions (chief among which were the idea that the break-in was staged, and the idea that Knox's "see you later" text meant she was confirming plans for a physical meeting that night), and that by the small hours of the morning of 6th November 2007 they genuinely, sincerely thought they'd solved the crime and caught the right people.
And IMO once they'd reached that point, another level of malpractice and incompetence kicked in: the type of confirmation bias that led the PM and police - who IMO were certain they'd figured out the crime and its protagonists entirely accurately, remember - to go out looking for evidence to support this conclusion. After all (in their heads), since they'd solved the crime properly, it ought to be fairly easy to find evidence of something which they were already convinced had actually happened (i.e. that Knox had met with Lumumba, and that the two of them (with or without Sollecito in those early days of the investigation) had gone to the cottage where Lumumba (with or without Sollecito) had attacked, assaulted and killed Kercher, and Knox and Sollecito had cleaned up the crime scene later that night or the next morning and staged the break-in). The police and PM already "knew" that all this had actually taken place, so they went looking for evidence of it - just as if you know a ship has sunk, you go looking for ship wreckage on the bottom of the ocean.
So in the case of Lumumba's bar for example, the PM/police had decided that Lumumba was a direct participant in the murder. And they knew that Kercher hadn't arrived back until around 9pm, and believed that the murder had occurred at something around 10.30-11pm at that point. In order for all this to be true (which they were sure it was), it would have been necessary for Lumumba's bar to have been closed for at least a large part of that evening. And since the police "knew" Lumumba was involved, they also "knew" by extension that his bar HAD to have been closed for a good portion of that evening. So they went looking for evidence of the thing they already KNEW to be true - that Lumumba's bar was closed for a good part of that evening. Naturally when they found someone who was prepared to agree with their position, they used that as confirmation of the "truth". I am guessing that they might have talked with others who'd told them that Lumumba's bar had in fact been open all that evening, with Lumumba present - but the police and PM would have written off such people as unreliable or somehow wrong in some other way, since the police/PM already KNEW that Lumumba's bar had to have been closed for a large part of that evening.
That's how I think this all went down: false (but sincere) certainty in the "truth" of a set of events and protagonists, based on erroneous inferences and other factors, followed by an improper confirmation-bias-led approach to finding evidence to "confirm" this already-decided "truth".