One needs to pay it off with cash or have another lender lined up .. 

Purchase price: $130,000 Cash invested: $20,000 Didn't expect my 2 bedroom rental to rent for what it did.

Any mistake can create a domino effect and impact both investments.If your current investment is performing well, save up that money and use it to build a pile of cash for a new investment.

I'm looking to purchase my first official investment property, but I'm running into steep competition against cash buyers. 

This presumes you make smart purchases that cash flow after paying all their expenses (Principle, interest, Taxes, Insurance, and a maintenance / Capex reserves).The risk of leverage is if your properties AREN'T making money - maybe you misjudged your income, or you run into repairs that cost more than you expected or budgeted for.  

@Sean Spagnola we work with a lot of real estate investors that provide us with intel on who the great property managers are.

I recommend the Midwest for its affordability and strong cash flow potential.

So, if things do go sideways, it's much harder to negotiate your way through.And, provided your loan is structured properly and closed with a competent title agent, I see no greater risk working with a private lender than with a traditional one.

They claimed this was not their fault as they did not provide the washer, but they provide the washer and dryer hookups and the pipes that regularly leak into the basement, garage, and kitchen.

Who is your competition and how will you provide a competitive advantage over them?