The following two papers talk about the subject; you may want to check them out:
Credit information sharing and loan default in developing countries: the moderating effect of banking market concentration and national governance quality
S Fosu, A Danso, H Agyei-Boapeah, CG Ntim… - Review of Quantitative …, 2020 - Springer
Bank size, market concentration, and bank earnings volatility in the US
J De Haan, T Poghosyan - Journal of International Financial Markets …, 2012 - Elsevier
This is a very interesting research question which implicitly touches four economic elements, viz:
(1) Market Concentration,
(2) Banks,
(3)Bank Defaults, and
(4) The Business Cycle.
Before we specifically link them we must understand :
(a) Market Concentration can exist even within an economy which is not served by any type of banking system. Market concentration is basically closely related to monopolistic situations, trusts concentratioon, and other structures where is prevalent market domination. All governments which, gratefully, have legislated against situations of market dominance in their jurisdictions, have mostly done so not because of any lamentations or otherwise in the banking sector.
(b) Counterintuitively banks exist and will exist in economies which both have or do not have situations of market dominance in their economies. Banks are there to serve whatever and whicever customers may exist, and if, in any specific economic sector, there happens to be one or more market concentrations they normally do not feel reticence against serving such concentrations too in what may happen to be their banking needs.
(c) It follows from (c) above that banks normally like to operate in economic structures where they are serving more - and not just one - customer (concentration) or sectors. Consider this particularly in the case of small economies. Most small economies which I have studied closely suggest to me that they view marketr concentrations with great reticence and structures (legal and otherwise) to quell such dominant operators and their operations. As you would know there are indeed many factors that may lead to market default, and situations of market dominance in any one specific sector is only one of them. The "caeteris paribus" (everything else remaining equal) reasoning is importabt here. A single bank solely dependent on one customer for the bulk of its profits (i.e. only having a single market concentration as its only (or main) customer is, caeteris paribus, yes more likely to face default than a comparable bank that has many customers.
(d) The jury verdict on market concentrations, bank defaults, and business cycle, is still not totally out yet. Research carried out by some of my undergraduates has so far elicited that there is a negative relationship between situations of market concentration and both high and low points in any business cycle. The business cycle of course effects both banks and other economic sectors. Banks will, in the downside of a particular cycle, perform no differently than other sectors; similarly in an upside part of it.
But, as I imply, it would be wrong to just say that there would always be more or less bank defaults in an economy with market concentration during any specific points of the business cycle.
I would suggest referring the following article "The impact of macroeconomic factors on risks in the banking sector: a cross-country empirical assessment"
The simple answer is of course "Yes". Market concentration, and the business cycle, are economic realities which however both have their own dynamic, viZ that at times they work very much independently of each other, and at times are effectively impacted by each other. One only has to study in depth Kondratieff's analysis of long wave economic and trade cycles to see how over history there were in fact times when the business cycle moved into an upward conjuncture even as, in certain selected countries, market concentration had not yet been subjected to controls, anti-trust regulating, etc.
And then there are periods when the reverse was in place.
Then, in the second overall scenario, there were times when the business cycle functioned in a certaiun conjunctural manner, but this was totally unaffected by any presence of market concentrations.