TL;DR: Maybe, but I don't think we can tell you.
This probably cannot be answered, at least fully. It depends if you want to think like a woodworker acting as an engineer, or an engineer acting as a woodworker.
Can a specific construction support ~160lbs? It depends on what you mean by "support", because it depends on what forces are applied.
What is this thing we call "Load"?
Something can be said to "support" a given static load if that static load can be carried through the members all the way to the ground underneath it. That is, you have to be sure that the wood or metal members can transfer that weight to the floor without major deflection or sudden or eventual failure. You also have to be sure your floor can handle these concentrated loads at whatever number points of contact the cabinet has. The whole thing is a system designed to transfer that load to the ground under your house.
And even if the construction could manage the dead-weight of 160lbs (or, more likely, 240lbs or more because you should give yourself at least 50% margin of error) you have to ask what happens when a hip hits it, or cat jumps on it, or there is an earthquake, or an alien invasion.
Think of a large mass at the top of a construction as an upside-down pendulum. Once you put energy into that mass it will want to move, and continue to move. This will stress the construction more than just transferring a static load to the floor, as joinery and fasteners and material will be stressed possibly beyond their limits, possibly leading to wear failure. You can't stop all of this movement, but you can restrict the amount of unexpected movement along any number of axes.
How do we even build things, anyway?
So you have to consider two major design points:
- Can the down- and cross-members handle the static load, transferring this load to the members below it to the floor? (And can your floor take this load?)
- Is the construction sufficiently stiff to keep dynamic loads from racking the construction, possibly leading to a catastrophic failure mode?
Woodworkers would probably look at this as a sort of robust table or cabinet construction. Building for 250lbs means you can't just take the same old china or microwave cabinet design without beefing up all the members related to transferring those loads. Joinery will have to be added to stiffen the "boxes" as well, or some sort of bracing will have to be added. You'd probably minimize "stretcher" distances without support, and use more robust lumber used in the typical ways to maximize the ability of wood to deflect more or less in different directions. i.e., now we especially care that dimension lumber is strongest with forces applied to edges (as in how joists are used) or in direct compression (as posts are used), and keep your spans under some distance that feels right.
An engineer would consult with the well-known tables and charts related to material failures and support limits and then tell you that you probably ought to at least have someone look at your floor joists before you do anything else.
This probably isn't rocket surgery
At the end of the day: can you (and maybe a friend) stand on the top of this cabinet and move around on it with minimal deflection, unexpected racking, or ominous cracking? If not, then you have your answer.
Are any of the load-bearing devices in the cabinet relying purely on the shear strength of metal fasteners? Are any of the forces wanting to pull fasteners out of mismatched butt-joints? If so, then you have your answer.
Here is the cabinet
