Are there algorithms with proven upper bounds but no proven lower bound (above constant time)? One of my professors mentioned such algorithms exist but could not think of any offhand. Obviously any algorithm will be at least $\mathcal{O(1)}$, but are there algorithms not yet proven to have a lower bound greater than this, which do have a proven upper bound?

I'm interested in both problems and algorithms, but in either case practical examples (if they exist) as opposed to artificial ones.
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!