Why shouldn't he? His brain, if we were to hazard a guess, is as capable as any other human and given his regenerative capabilities, his memories should be perhaps a bit more durable than most humans.
We are lead to believe his mutant power will allow him to repair any damage done to his body, this would include the very difficult to replace brain cells, which once damaged in normal humans, rarely or very slowly recover.
For Logan, it should be far easier. Logan should lose memories only if his mind is tampered with or he suffers catastrophic injury, and even then traces of his memory should still be available even if they are no more than dreams to him.
The brain is a fantastic organ
The human brain is one of the most studied and least understood organ in the human body. It is the seat of consciousness, the director of human volition, the palace of memory and believed to be the ultimate arbiter in what makes us what we are as a species. The process by which memory is enacted in the human creature is still one not completely understood.
Memory comes in a variety of types as well, sensory, short-term and long-term memories. Each is stored differently and can affect the others over time, weaving connections between all three. Each type of memory has several processes where memory is encoded, stored and later accessed. Wikipedia: Memory
Part of the process that is memory is based in a physical construction of dendrites, the connections between brain cells by which all activity in the brain is based upon.
But there are also neurochemical interactions taking place, serotonin, oxytocin and other such neurotransmitters which are believed to make the effective creation and activation of memory possible.
In recent years, scientists have begun to realize the brain is not just capable of memory in specific locations of the brain but that memory is a whole brain event, spreading memory across vast neural connections to entirely different regions of the brain.
It is possible that memory can be related to a variety of sensory conditions including sounds, smells, and tastes. Memories can be keyed to a particular event, trauma, or even the smell of a peach cobbler on a summer day on your grandmother's farm.
Wolverine's mind is an aspect of his brain and his mutation.
There could be any number of mitigating circumstances that could affect his memory including:
His age - If there was any single element I think that would affect his memory I would suggest his age is first and foremost. Since he is well over a hundred, we have no idea how memory is layered in his brain. It may be his earliest memories could be his least vivid or his most. Considering his traumas suffered at the hands of his father, it is possible those memories may drive his behavior even though he no longer remembers them specifically.
Physical trauma - Considering his lifestyle of physical stresses, combat and profound injury possible in his line of work, it is fortunate Wolverine's body regenerates from tissue damage. What we don't know is how complete the regeneration is. Do specific dendrites return to their previous configurations in his brain and if so, will their memories be retained?
Col. Stryker (from the movies) didn't think so, hence the reason he made adamantium bullets in order to scramble Logan's brain. Logan did appear to lose his short term memory once he had regenerated. Even then, he did seem to recognize the woman who used her powers to affect his memories at Three Mile Island.
On the other hand, Charles Xavier seem to think it was possible for those memories to be retained even if Logan couldn't specifically remember them without telepathic help.
Potent events - In the case of potent events like Hiroshima, it is possible for Logan to have memories of the event because it was so catastrophic as to be almost unbelieveable (and thus potentially unforgettable). We have survivors of the bombings of Hiroshima and Nagasaki who are very old now and can still remember in vivid detail what happened that day and every day after that.
Other mutations acting on Logan's mind - Stryker did have Logan teamed up with a mutant whose power was a touch-based hypnosis (In the first Wolverine movie) but unless she tampered with those specific memories, it is possible those memories remained available but obscured through the nature of Logan's vast experiences.
- Since Logan's brain is capable of putting itself back together again even after immense trauma, it is theoretically possible for Logan to remember things UNLESS they are specifically forced out of his mind by a more powerful or subtle mutation. This is what Charles Xavier did to help Logan remember his past even if Logan himself had repressed those memories or they had been lost to layering or age.