Is there inverse relationship betweet prices of interest-bearing assets and the interest rate? I was wondering about why investors converting out of interest-bearing assets into money in the money market cause the interest rate to increase. Then I remembered that there is inverse relationship between prices of bonds and the interest rate. In case of bonds investors converting out of them would increase their supply, consequently lowering their prices. But lower prices of bonds mean higher de-facto interest rate for your investment into bonds because the quantity of money that you needed to make investment has decreased. Then it happened to me that the same logic can apply to interest-bearing assets in the general, not just to the bonds. And if it's correct it can serve as an explanation for exact mechanism of for increasing interest rate in the money market.
Am I correct about the relationship? Is there inverse relationship between interest-bearing assets and the interest rate? And if not, why?
The shortest possible answer to the question in the title (as stated now): “Yes.”
However, based on the comments, there are questions about definitions. I will attempt to answer them.
A fixed income instrument is anything that pays the owner cashflows on a fixed legal schedule, and the owner is not obligated to make any payments (other than the initial payment). This covers bonds, loans, and other forms of debt.
If there is no optionality in the structure (not prepayable, etc.), the internal rate of return of the security is unique and has an inverse relationship with the initial price. (Excluding corner cases such as zero cost, etc.)
This internal rate of return is typically referred to as the rate of interest or yield of the instrument. (Bond and money markets have specific calculation conventions, which end up being a more complex version of the internal rate of return.)
When trading fixed income securities, they can be traded in terms of yield or price interchangeably (although markets follow conventions for which to use in a quote). As such, prices and “interest rates” are mechanically always moving inversely.
The question seems to be focussed more on the policy rate, which is often “the” interest rate in simplistic economic models. Modern central banks force a short-term nominal instrument rate (often overnight, but can be two weeks or even three months) to some target level, that is changed over time as a policy tool.
Despite what simplistic models say, there is no relationship between the “money supply” and this rate. (People discussing this relationship are typically using Economics 101 textbooks that are decades out of date.)
The yields on other instruments move around on a daily basis. Many factors can influence prices, but there are bedrock valuation principles.

Instruments that have no perceived default risk (e.g. most non-Euro area developed country sovereigns) have a fair value that matches the expected path of short-term rates. The set out of outstanding bonds defines a risk-free yield curve.
Other instruments will incorporate a default risk premium and/or a liquidity premium that is added to the risk-free yield curve.