ATB Sales, owners of cycle maker Whyte Bikes, had sought an injunction in the London High Court against Rich Energy -- a little-known energy drink brand that became Haas's sponsors this season -- to require them to remove the logo.
Whyte argued that Rich had copied a logo belonging to the cycle brand that was designed by company employees in 2008.
Judge Melissa Clarke said in a written judgement that she found Rich Energy boss William Storey and Sean Kelly, founder of the digital marketing company Staxoweb Ltd, to be "poor witnesses".
"He had a tendency to make impressive statements, which on further investigation or consideration were not quite what they seemed," she said of Storey.
The judge cited as an example that when cross-examined about a claim to the media in February that 90 million cans had been produced, Storey explained that Rich "had produced 90 million cans, but had not yet filled and sold them".
Rich Energy said on their Twitter feed that they were disappointed with the judgment and the findings of the judge.
"We are considering all of our legal options including appeal," they added.
Rich Energy tried to buy Force India (now Racing Point), to some scepticism from F1 insiders when that team went into administration last year. They were also in talks with struggling former champions Williams.
Storey shrugged off those who questioned the company's fundamentals at the Haas team launch in February and said Rich had a presence in 43 countries.
Haas's drivers are Frenchman Romain Grosjean and Denmark's Kevin Magnussen.
(Reporting by Alan Baldwin, editing by Nick Mulvenney)
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