Are there algorithms with proven upper bounds but no proven lower bound (above constant time)?
To try and answer your question:

Yes. generally lower bound is much harder to prove than upper bound.
That is, to prove an upper bound, you must only show one algorithm that solves problem $A$ in $t_1(n)$ time.

To prove a lower bound $t_2(n)$ for the same problem $A$, you must show that ANY algorithm that solves $A$ MUST work in at least $t_2(n)$ time.

Take the median for example (under the comparison model). There is a recursive algorithm that solves the median problem in $O(n)$ time. That means any greater upper bound is moot, and any greater lower bound is wrong. 

However a precise lower bound for the median is an open question. The algorithm mentioned above performs $c*n$ comparisons, where $c$ ~ $17$. It has been proven that for selection of median, a total of $\Omega(\frac{3n}{2})$ comparisons is required (which means a better algorithm does not exist).

However, we don't know if a greater lower bound exist. Proving a $\frac{5n}{2}$ comparisons lower bound is an open question!