IBM's quantum experience offers computing up to 15 qubits however through my search of current literature the largest implementation for the algorithm has been with 4 qubits. Is there an issue, preventing people from increasing the size of the algorithm?
Welcome to the quantum computing stack exchange, Grover's algorithm is defined for an infinite number of qubits. You need $n \approx \log_2 (N)$ qubits where $N$ is the number of items searched, so the number of qubits needed depends on the number of items changed. Pay attention you need to round up.
The problem is, a physical quantum computer needs to have a property called connectivity. Connectivity is the ability for qubits to have a link and therefor be in superposition (comment if you need help with the terminology). Since all the qubits used for searching with Grover's Algorithm have to be in a superposition, you need the maximum connectivity to be able to perform the algorithm.
I am in no means an expert on the subject, but I think the higher the connectivity, the harder it is to keep the state coherent (this means making sure that no noisy errors happen). Therefor, the algorithm may not be performed on larger quantum computers (with $\approx$30 qubits), since they do not have the required connectivity.
However, I know there exist quantum computers with 15 qubits and maximum connectivity, though maybe they have other technical restrictions. I am pretty sure that Grover's Algorithm was executed on a larger number of qubits, so I would consider doing a bit more research. In any case, you can always simulate the program on your own computer like languages like Cirq, Qiskit and Q#.
If you have any other questions, please ask !