Does treasury bill have opening and closing dates?

Yes. In Nigeria, Treasury Bills (T-Bills) are sold through periodic auctions conducted by the Central Bank of Nigeria, so they are not available for purchase every day.
How Treasury Bills Work
The CBN announces an auction date and the tenors available (typically 91-day, 182-day, and 364-day bills).
Banks collect applications from customers before the auction date.
Once the subscription window closes, you cannot buy that particular issue anymore.
You then have to wait for the next auction.
This is likely why your bank told you:
“Come back in two weeks” (waiting for the next auction window).
Then later “It’s closed” (the application deadline for that auction had passed).
Why This Happens
Some banks:
Stop accepting applications a few days before the auction.
Have internal cut-off times that are earlier than the CBN’s deadline.
May prioritize larger subscriptions when demand is high.
How to Avoid Missing Out
Ask your bank specifically:
When is the next Treasury Bill auction?
What is their application deadline?
Submit your instruction and funds before the bank’s cut-off date.
Consider investment platforms that provide Treasury Bill access and notify users of upcoming auctions.
Alternative While Waiting
If your goal is short-term capital preservation and earning interest, a Money Market Mutual Fund (MMF) can be a good temporary parking place for your funds because:
You can invest any time.
There is no auction window.
Your money remains relatively liquid.
Current yields are often competitive with short-term fixed-income instruments, though not guaranteed.