Your professors are raising an important point. Unfortunately the English usage is such that I'm not absolutely sure what it is they said. Let me answer the question in terms of non-toy programs that have certain memory usage characteristics, and that I have personally worked with.
Some programs behave nicely. They allocate memory in waves: lots of small or medium-sized allocations followed by lots of frees, in repeating cycles. In these programs typical memory allocators do rather well. They coalesce freed blocks and at the end of a wave most of the free memory is in large contiguous chunks. These programs are quite rare.
Most programs behave badly. They allocate and deallocate memory more or less randomly, in a variety of sizes from very small to very large, and they retain a high usage of allocated blocks. In these programs the ability to coalesce blocks is limited and over time they finish up with the memory highly fragmented and relatively non-contiguous. If the total memory usage exceeds about 1.5GB in a 32-bit memory space, and there are allocations of (say) 10MB or more, eventually one of the large allocations will fail. These programs are common.
Other programs free little or no memory, until they stop. They progressively allocate memory while running, freeing only small quantities, and then stop, at which time all memory is freed. A compiler is like this. So is a VM. For example, the .NET CLR runtime, itself written in C++, probably never frees any memory. Why should it?
And that is the final answer. In those cases where the program is sufficiently heavy in memory usage, then managing memory using malloc and free is not a sufficient answer to the problem. Unless you are lucky enough to be dealing with a well-behaved program, you will need to design one or more custom memory allocators that pre-allocate big chunks of memory and then sub-allocate according to a strategy of your choice. You may not use free at all, except when the program stops.
Without knowing exactly what your professors said, for truly production scale programs I would probably come out on their side.
EDIT
I'll have one go at answering some of the criticisms. Obviously SO is not a good place for posts of this kind. Just to be clear: I have around 30 years experience writing this kind of software, including a couple of compilers. I have no academic references, just my own bruises. I can't help feeling the criticisms come from people with far narrower and shorter experience.
I'll repeat my key message: balancing malloc and free is not a sufficient solution to large scale memory allocation in real programs. Block coalescing is normal, and buys time, but it's not enough. You need serious, clever memory allocators, which tend to grab memory in chunks (using malloc or whatever) and free rarely. This is probably the message OP's professors had in mind, which he misunderstood.