Enough time to make plane connection?

If your luggage will go all the way to San Francisco and you will not have to recheck it in Frankfurt and your plane from Dubrovnik is not delayed, one hour should be enough time.

We lived this several years ago. Although we had what looked like a 2 hour layover, the flight to Frankfurt was delayed an hour. We landed with about an hour to go, only to encounter a very long line for passport control. Now, running to the gate we encounter another full security screening. We are now hearing our name paged. They see something in my daughters luggage they don't like (tea from Bosnia). A flight attendant shows up yelling for us to hurry or we will miss the plane. Nothing we can do but sweat. We clear security running for the gate, with flight attendants yelling "hurry, hurry!". Yes, we make the flight. Long way of saying, it will be tight, and pray the flight from Dubrovnik is not delayed. Our tickets were booked as one so I think that explains the extra efforts to get us aboard our fight to SFO; Lufthansa by the way.

In the homeward bound direction, minimizing carry-on to must have/irreplaceable stuff and checking as much as you can simplifies connections. First of all, you can move quicker with a lighter, less bulky carry-on load. And as an earlier respondent notes, minimizing the likelihood of rescreening and/or hand searching your carry-on luggage reduces your loss time risk. And if you checked luggage doesn't make the connection, the airline will deliver and you don't need to be a pack mule carrying your luggage home.

I did this tour a couple of years ago. I used the same route on the way home. I elected to spend the night in Frankfurt to avoid the mad dash, worrying etc.

My rule of thumb is to allow no less then 2 hours 45 minutes to change planes.
I violated this in my initial trip planning recently ( 2 hours at Gatwick, smaller airport, no problem ...) and my first flight was changed to 25 minutes later ... so, I had to change my flight to 1 day earlier, pay a hefty change fee, and pay for an over-night at Gatwick.
So I would caution NO!
If you are booked on one airline as a through flight, maybe ... since they then would be obligated to get you on the next available flight to SFO. But, depending on the time of day, that might be a day later anyway.
I wouldn't risk it. Why set yourself up for stress at the end of a terrific adventure?