In short: Fuck no!
Now that I have the vulgarity out of the way, it is time to take that question and examine it.
A person involved in any creative craft will often ask themselves that very question, just taking the “writer” and replacing it with said craft. For example, the question could be “Am I a good painter?” Or, “am I a good sculptor?” The point is that many artists or creative types such as me ask themselves this question time and time again.
In relative terms, there are technical aspects to any art that can be used to measure an individual in how they perform. It can be in the knowledge of what tools to use. Experience in using said tools. There could also be an argument in the catalog of knowledge one might have in a particular field lends to whether or not someone is “good.” An example would be a painter who has an extensive knowledge base on the varying styles in painting vs one who may be very narrow in focus and knowledge.
Am I proficient with grammar?
Can I diagram a sentence?
Can I build a basic plot?
Etc, etc, etc. When examining skills, we all fall somewhere on what is known as a bell curve. The bulk of the population of writers in this case fall between two ends of the spectrum: experts in the technical aspects of writing and those who are just a complete mess of wordsmithing. This goes for either end: nonfiction or fiction writing.
Now I have no idea where I fall on the curve. I can be sure that it is somewhere in the middle. Which way I lean? No idea. What I believe vs truth is that I lean towards the “just a complete mess of wordsmithing.”
But there is another element to whether someone is a good writer (artist) or not. It is the most important factor, and one that is entirely elusive. It is subjective. This is what I struggle with. I believe it is what every artist struggles with: their voice. It’s the style. It’s what brings people to appreciate the work no matter what the technical proficiencies are. It’s why some writers may be pop culture hits while others stand the test of time. It’s a skill that is grown. And while some are born with it, others cultivate it.
So am I a good writer? Who knows. But I hope to be some day.