I am not sure what the proper definition of a serious Bible student is; but, for sure, anyone who claims to be an instructor in righteousness based on the Bible should self identify as such. That would naturally include any individual who practices Christian evangelism, be they clergy or layman.
For my money, the serious Bible student is not known by the questions he or she can answer, but rather from the many questions and unresolved issues related to God's nature, plan, and mode of operation the student has identified and still struggles with. Coupled with the list of unanswered questions, I would be most impressed by the number of significant changes in understanding the student has evolved through, in the course of personal Bible study.
I ask myself a series of pertinent questions as a test of my Bible knowledge. What does understanding the Bible really mean- I know what it says, or I know what it means when it says it? How much of the Bible do I comprehend? 10%, 50%, 85%? How much of what I claim to understand do I actually understand correctly? 100%, 75%, 50%? In order to be an effective evangelical instructor in righteousness, how much of the Bible would I need to understand 100% correctly? What is the basis for my certainty around my personal understanding of the portion of the Bible I claim to know well enough to qualify as an instructor? Is that certainty based on what I have been taught by others who are supposedly wiser in the Word? Is it based on a reliance on church tradition? Has my certitude evolved over time because of seriously pondering the big Bible questions and wrestling with all the associated issues and implications?
If serious Bible study is actually marked by a significant growth in understanding, which implies fundamental change, then it is hard to conceive how the average evangelical could qualify. For such traditionalists, theological change is anathema. One cannot meaningfully seek a deeper understanding of sacred Truth which was purportedly deciphered and codified over 1000 years ago. All one can do is embrace and regurgitate such truth without question. If questioning is forbidden, then serious study is also.