Yes, they can.
In fact, they’re much better climbers than most people realise.
I’ve found bed bugs behind pictures, inside curtain folds, around light switches and even hiding behind loose wallpaper.
They aren’t confined to the bed.
Why would they climb a wall?
Because walls often lead to good hiding places.
A small crack behind a skirting board or a gap around a socket provides exactly the sort of shelter bed bugs like.
They’re simply looking for somewhere dark, narrow and undisturbed until it’s time to feed again.

Does this mean the infestation is severe?
Not necessarily.
Finding one on a wall doesn’t automatically mean the infestation is out of control.
It simply tells me they’ve moved away from the bed.
The next step is to work backwards and find where they’re actually living.
One thing I’ve noticed over the years
People often spend hours inspecting every wall in the bedroom before they’ve properly checked the bed.
I’d always do the opposite.
The bed, headboard and frame usually tell me far more than the walls ever will.
My advice
If you’ve found a bed bug climbing a wall, don’t panic.
Treat it as a clue rather than the answer.
A careful inspection around the bed will usually reveal much more.